So, the short answer is, yes, he was involved.

I have a literary background rather than being a historian by trade, so forgive me if I am approaching this wrongly, but I believe I can answer some of your questions.

One thing to note is that we really don't know anything about Shakespeare's activities between 1585 (the birth of his son is recorded) and 1592, when he's mentioned rather scornfully by Robert Greene, a playwright, in the tract, Groat's Worth of Wit'. So in these years between it's entirely possible he had work scorned or rejected. By the mid 1590's though he was quite established as a popular playwright - popular enough to start having his work published and to be attacked in print form by another playwright! It's doubtful that he was suffering much rejection at this point.

In terms of how he got his work performed/published, from 1594 he was writing for the Lord Chamberlain's Men (later the King's Men when James 1 took to the throne), a newly established theatre company in London. They were based first at the Theatre and the Curtain, then moved to the newly built Globe in 1599. So he wasn't submitting his plays for approval with a publisher or sending in scripts to different theatres.

It's hard to be precise about what plays Shakespeare himself acted in, but as a sharer (think shareholder, basically) in the Lord Chamberlain's Men, he would have done some acting. He's also listed as an actor in a list of Shakespearian actors in the First Folio edition. However, to the best of my knowledge, the folio does not list what roles he played. of course, the most famous actors of the Lord Chamberlain's Men were Richard Burbage and the comic actor William Kempe, so it's unlikely Shakespeare was playing many leads. In the folio version of Ben Jonson's 'Every Man in His Humor', Shakespeare is listed as playing the part of Kno'well, one of the leads.

So, I've already mentioned folios, which leads me onto answering your question about publishing. Our earliest copies of Shakespeare's plays, produced in his lifetime, come in the form of cheap quartos. The first was Titus Andronicus, published in 1594. His works weren't collected together until the First Folio, published after his death. There's a lot of changes between different editions of his plays. For instance, that most famous of soliloquys, "To be or not to be..." changes quite a lot, depending on which text you read. The first quarto version, from 1603, is vastly different from the 1604 quarto edition, and the First Folio version. Now, it could be the first quarto was a so-called "bad quarto" - a cheap knockoff, or it could merely be a rewrite. The fact the First Folio wasn't produced until after Shakespeare's death also throws up the potential for revisions and rewrites by a hand other than Shakespeare's own. It's important to bear this changing nature of Shakespeare's work in mind: these were plays written to be performed, and so rewrites and revised drafts are to be expected.