Did Shakespeare really "invent" all of the words and phrases he purportedly did, or did his work just contain the first written record of them?

That's a great question. It's a great question because it has no definite answer. He was clearly obsessed with words and it is widely presumed that he was an inventor as well as a collector because there are so many common English words that appear for the first time in his plays, poems, and sonnets. However it is unlikely that he invented all of them.

The following is from an answer I gave to a related question a few months ago:

Often the "new" words were combinations of multiple preexisting words like eyeball (first appears in The Tempest Act 1, Scene 2, Line 302), bedroom (A Midsummer Night's Dream 2.2.51), moonbeam (Midsummer 3.1.160), worthless (Henry VI, Part III 1.1.102), and bloodstained (Titus Andronicus 2.3.211). Other times he would take existing words and add syllables or change their parts of speech. The preexisting word assassin became the Shakespearean word assassination (Macbeth 1.7.2), meditate became premeditated (Henry VI Part I 3.1.1), etc.

The Complete Works of Shakespeare include at least 1,700 words that have no other literary precedent.

It is important to understand that at the time Shakespeare was writing, the rules, spelling, and grammar of the English language had yet to be codified. By the time that Modern English was being standardized in the late 17th and early 18th centuries, Shakespeare's works were widely popular & their contribution to what would become "official" English was massive. Samuel Johnson's A Dictionary of the English Language (1755) cites Shakespeare more times than any other writer in its definitions of words and phrases. His influence on the expansion of the English language as well as poetic and grammatical structures is very difficult to overstate.

Until the 14th Century, French was the official language of England due to Norman/French cultural dominance. English was not recognized as the official language until 1509 and up until 1583, the rhetoric of what would later become Modern English was primarily indebted to Chaucer. Other than Chaucer, the relative lack of written records from the period makes the innovation of our contemporary common tongue uncertain. According to literary critic Boris Ford:

Before the arrival of Shakespeare to London, there was little hope for the future of English but by 1613, when Shakespeare's last work was written, the literature of modern English was already rich in varied achievements, self confident and mature.

The scope of Shakespeare's vocabulary is simply mind-boggling. The King James Version of the Old Testament contains 10,867 unique words. The works of Milton use ~15,000 - 16,000. According to the Shakespeare Lexicon and Quotation Dictionary, The Bard employs an astonishing 27,780 unique words.

Some of these new words were his invention, others were simply written down for the first time by him. Alas, it is difficult (perhaps even impossible) to ever tell which is which.