I know that we know the names and probable roles of several actors in Shakespeare's troupe like Richard Burbage and William Kempe, and I know that Shakespeare himself acted in his own plays and plays by other authors presented at the Globe, but do we have any idea which roles he played?
It is unknown exactly which roles he played.
We can verify that Shakespeare did indeed act in his own plays and in those written by his contemporaries. The First Folio of 1623 names him first among the actors that played in the original productions. Playwright Robert Greene mentions Shakespeare as an actor as early as September of 1592 in his book Groat's-Worth of Wit.
In his 2000 book, Shakespeare, the Player, actor John Southworth (citing 19th Century Shakespeare Scholar Frederick Gard Fleay) makes a compelling argument that Shakespeare played the title role in Edward I by playwright George Peele in 1593. As for performing in his own plays, there have been multiple claims and speculations over the past four hundred years but all of them were written by scholars who lived at least one century after Shakespeare and the original sources that the claims are based on either no longer exist or were fabrications in the first place.
That said, the following roles are commonly proposed as having been played by Shakespeare himself in their first run:
King Duncan (MacBeth)
Adam (As You Like It)
King Henry (Henry IV, Parts 1 and 2)
The Ghost (Hamlet)
