This is a complicated question to answer -- I don't think it can be a simple yes or no.

There are certainly parts of Shakespeare's work that anyone would understand, then or now. Especially in the lighter comedies. When Helena says "use me but as your spaniel" in Midsummer while waggling her butt at Demetrius, everyone knows what she is referring to. This effect is amplified through intense poetic repetition that whole speech goes like this:

Helena-- I am your spaniel. And, Demetrius, The more you beat me, I will fawn on you. Use me but as your spaniel—spurn me, strike me, Neglect me, lose me. Only give me leave, Unworthy as I am, to follow you. What worser place can I beg in your love— And yet a place of high respect with me— Than to be usèd as you use your dog?

All the audience really needs to get from this speech is Helena loves Demetrius like a dog loves his master. There's nothing really very complicated there. This effect is further amplified by the actions onstage. Anyone who has seen this speech in a production will instantly understand it. Helena is fawning over Demetrius, desperately pretending to be a dog to gain his love. The action on stage would have made the meaning of the lines quite obvious.

Not all of Shakespeare's works are so easy to follow though. Take Titus Andronicus for example. The first act of Titus is extremely difficult to follow -- so many characters are introduced (a few who die immediately) and plot points fly by with a speed that makes it hard to believe that audiences could follow. While audiences might not remember that Alarbus is the name of the son of Tamora who Titus sacrifices to the Roman gods, they would certainly get the gist of the idea because Tamora repeatedly mentions the wrong done to her throughout the rest of the play. Titus Andronicus is a lot like Taken (the Liam Neeson movie), except the 10 minutes of exposition is actually 25 minutes and incredibly confusing. It wouldn't matter if Elizabethan audiences would have understood exactly what happened in the first act, because the repetition throughout the rest of the play makes it impossible to avoid -- and in the end Titus is all about blood and explosions. In fact, Titus Andronicus (often called his worst play) is the work that launched his career and was his most popular in the day.

This is a roundabout way of saying the answer is complicated. I think it would be naive to say that audiences of the time understood everything, but anyone who is really familiar with Shakespeare's work will tell you that the staging of a play and the way Shakespeare uses poetry (especially repetition) to reinforce plot points makes it very hard to leave the theatre without at least an idea of what was going on.

Sources: John Basil, artistic director of American Globe Theatre's Shakespeare lectures -- I've been producing, directing and acting in Shakespeare's work for 5+ years

Will in the World by Stephen Greenblatt might be a good read for you too. It doesn't answer this question per se, but it has a lot of information about contemporary literary criticism of his work.

The answer to your question is a big plate of "yes" with a small side dish of "mostly".

Emotion is the actor's currency. Language and motion are the mediums of exchange. Theatre is the act of presenting the experience of a real or imagined event before an audience. Without effective emotional and intellectual communication, a play can't fulfill its purpose.

Shakespeare's writings suggest that he cared passionately about presenting the audience with an experience that was as lifelike as possible. In Hamlet, the title character coaches a group of actors on how to conduct themselves during a performance. This has been widely interpreted as Shakespeare's own advice to actors about how his plays should be performed:

HAMLET:

Speak the speech, I pray you, as I pronounced it to you, trippingly on the tongue: but if you mouth it, as many of your players do, I had as lief the town-crier spoke my lines. Nor do not saw the air too much with your hand, thus, but use all gently; for in the very torrent, tempest, and, as I may say, the whirlwind of passion, you must acquire and beget a temperance that may give it smoothness. O, it offends me to the soul to hear a robustious periwig-pated fellow tear a passion to tatters, to very rags, to split the ears of the groundlings, who for the most part are capable of nothing but inexplicable dumbshows and noise: I would have such a fellow whipped for o'erdoing Termagant; it out-herods Herod: pray you, avoid it.

Be not too tame neither, but let your own discretion be your tutor: suit the action to the word, the word to the action; with this special observance: that you o'erstep not the modesty of nature: for any thing so overdone is from the purpose of playing, whose end, both at the first and now, was and is, to hold, as 'twere, the mirror up to nature; to show virtue her own feature, scorn her own image, and the very age and body of the time his form and pressure. Now this overdone, or come tardy off, though it make the unskilful laugh, cannot but make the judicious grieve; the censure of the which one must in your allowance o'erweigh a whole theatre of others. O, there be players that I have seen play, and heard others praise, and that highly, not to speak it profanely, that, neither having the accent of Christians nor the gait of Christian, pagan, nor man, have so strutted and bellowed that I have thought some of nature's journeymen had made men and not made them well, they imitated humanity so abominably.

O, reform it altogether. And let those that play your clowns speak no more than is set down for them; for there be of them that will themselves laugh, to set on some quantity of barren spectators to laugh too; though, in the mean time, some necessary question of the play be then to be considered: that's villanous, and shows a most pitiful ambition in the fool that uses it. Go, make you ready.

• Hamlet (Arden Shakespeare, Third Series) Act 3, Scene 2, Lines 1-45

If you prefer to listen rather than read it, here's David Tennant, or Kenneth Branagh, or Lawrence Olivier.

That is not to say that the dialogue was completely lifelike. Nobody went around in the streets of London in the 1590s speaking in blank verse and Shakespeare's vocabulary was remarkably large compared to the laity.

Performing blank verse is an example of what actors call using "heightened speech." It is a communicative and emotional midpoint between speaking and singing. When speaking in verse, the actor's dialogue is written out like a piece of music with punctuation acting as rests, capitalization as emphasis, and each word's syllabic count dictating the pace. It would have been understandable but clearly not a representation of day-to-day speech.

Hamlet's mention of the groundlings as "capable of nothing but inexplicable dumbshows and noise" speaks to an intellectual divide between the distinct classes of people who attended these plays. The people who could only afford standing-room tickets were catered to with bawdy humor and titillating sexual content, while the more worldly and educated were targeted with dramatic timing, soaring poetry, and the philosophical introspection of the characters.

There is also the matter of Shakespeare's "invented" words. The plays contain the first known use of more than 1,700 English words. Some of these might have gone over the heads of some audience members, but for the most part careful watching and listening would have been enough to convey the meaning of the words.

In short, the plays definitely made sense to their original audiences because they had to in order to produce effective storytelling.

For further reading/viewing:

Speaking Shakespeare by Patsy Rodenburg, 2002. Patsy Rodenburg is the Head of Voice at the Guildhall School of Music and Drama in London, the Director of Voice at Michael Howard Studios in New York City, and has also worked with the Royal Shakespeare Company and Royal National Theatre.

Playing Shakespeare series presented by John Barton, 1982. It's a little dated but this video series is still used all over the world to train actors to perform Shakespeare's words.

This question may have been answered to your satisfaction but i wanted to add a couple of more thoughts on the differences between Shakespearean audiences and contemporary readers of Shakespeare:

The most obvious one is that languages grow and change over time. Modern English is somewhat different from Early Modern English (Shakespeare's English, or Elizabethan English). The biggest grammatical difference is the use of thee and thou as a grammatical person (along with the -t and -st verb endings that went with it, such as "goest" or "shalt"), but another one is EME's usage of the subjunctive mood ("if i were rich" vs. "if i was rich"). now if both of those sound grammatical to you it's because they are -- in other words we still use the subjunctive mood, just not as often as it was used in EME, where it was frequently employed in both independent and subordinate clauses. The same goes for the use of the infinitive (Shakespeare's line "too proud to be so valiant" is more easily understood by the ME speaker as "proud of being so valiant"). This is a baseline Shakespeare had with his audience, who would all have been EME speakers, of course. So up until now, the answer is yes.

Then we run into the problem of verse, which it is important to note, not all of Shakespeare's plays are written in. IIRC, Much Ado About Nothing is written mostly in prose. But verse was also a theatrical convention of the time, as was heightened or poetic language (see: Marlowe's Tamburlaine or Kyd's Spanish Tragedy). So it makes a certain amount of sense that at the very least theatergoers understood and accepted the convention. So far, the answer is again yes.

Then we have to start looking at Shakespeare's own innovations or adaptations of current convention. Although word invention was pretty common during the Elizabethan era (in that playwrights that were not just Shakespeare were inventing words at a comparable rate to Shakespeare) Shakespeare took many grammatical liberties in his works that other playwrights, such as Ben Johnson or Thomas Middleton, generally did not. [EDIT: i can only assume this was done more than anything to fit a verse line or for dramatic/comedic reasons, but that's literary conjecture.] Examples include switching the standard subject-object-verb (SVO) syntax to SOV ("Thou has thy father much offended", Hamlet) or OSV ("These strong Egyptian fetters I must break", Anthony and Cleopatra), as well substituting nouns for verbs and vice versa ("This day shall gentle his condition", Henry 5), omission of relative pronouns ("I have a brother is condemned to die", Measure for Measure), etc. The list is really, really long. Back to your question: did people understand this part of Shakespeare's writing? That I don't know, but I suspect the answer would be yes. Why? Well, probably for the same reason that contemporary audiences can understand Yoda in Star Wars: while the grammar is often inverted, it is only done to a degree where meaning is not completely lost, partly thanks to the context in which the words are spoken.

Which brings me to my last note. Shakespeare's work was meant to be spoken, not read. I would argue that any remaining difficulty in understanding Shakespeare language would be removed for the audience member (either a ME or EME speaker) if it was spoken by an actor who, with gesture and vocal inflection, could illuminate whatever is unclear. This is particularly true of jokes, puns, and emotionally or philosophically dense material -- it is more easily understood when it is conveyed by a human and not by a page. The testament to this premise is in the sage old advice given to people approaching Shakespeare for the first time: "Don't read it, go see it." Given the increasing popularity of free outdoor or even prison Shakespeare performances, I would say most people continue to be able to understand his language enough to be able to access his plays. That of course, is a also bit of conjecture on my part though :)

On a related note, would the average 16/17 century Englishman be going to a Shakespears play, or were the audiences the upper 1% only?

Best I can find elsewhere on Shakespeare and his audiences going by quotes in his plays describing groundings (so cheaper average ticket goers):

O, it offends me to the soul to hear a robustious periwig-pated fellow tear a passion to tatters, to very rags [referring to bad stage acting], to split the ears of the groundlings, who for the most part are capable of nothing but inexplicable dumbshows and noise

.......Now this overdone, or come tardy off, though it make the unskilful laugh, cannot but make the judicious grieve, the censure of the which one must in your allowance o'erweigh a whole theatre of others

.......And let those that play your clowns speak no more than is set down for them, for there be of them that will themselves laugh, to set on some quantity of barren spectators to laugh too, though in the meantime some necessary question of the play be then to be considered. That's villainous, and shows a most pitiful ambition in the fool that uses it. Go make you ready. (Hamlet, 3.2)

These are the youths that thunder at a playhouse, and fight for bitten apples; that no audience, but the Tribulation of Tower-hill, or the Limbs of Limehouse, their dear brothers, are able to endure. (Henry VIII, 5.4.65-8)

Doesn't describe what the average person thought, but to Shakespeare they were at least noisy and dumb and mostly there to laugh.

Does anyone know if it's like old musical hall theatre? where average or groundling might have gone not for enlightenment of a good story, but the opportunity to be entertained either by the jokes in the play or to throw stuff at the actors, heckle and gather with friends at a local show. It does seem like Shakespeare uses comical characters or some violence to keep groundings on side, if they didn't understand the play or language then the heckles would take over the show. I know in Henry V it was common for the actors playing the French characters to get items thrown at them from the pits like it was part of the play, but by this theory best left to the experts.

Since most major plot lines were explained three different ways by different characters (complicated with literary allusions, metaphorically, and in plain language) short answer is YES. I taught Hamlet for 10 years to 10th graders and I answered this question a lot and was able to show literal proof throughout the play at many points.