What does Shakespeare mean by the word 'a?
It usually means "he". Thus the Nurse in Romeo and Juliet: "God be with his soul! 'A was a merry man!"
It's an archaic abbreviation of sorts. Pronounce it "uh".
Please note that the apostrophe precedes the word: it's 'a not a'.
Shakespeare wrote in English, the same language I am using now. There is no such language as "Shakespearean language" or "Shakespeare language". It's English. A word like "then" is a building block of the English language and always means "then" when Shakespeare or any other English speaker uses it.
Shakespeare uses the word "bug" to mean a bugbear (this is a word Shakespeare also uses), a spook, a bogeyman (a word which derives from "bug"), something to frighten children. In a famous early-sixteenth century version of the Bible, Ps. 91 is translated "Thou shalt not need to be afraid for any bugs by night." The KJV substituted the word "terror". Shakespeare uses this word only five times, perhaps most characteristically in A Winter's Tale…
Well, I am not sure what you mean by "Shakespeare words". If you mean words that Shakespeare used, "What are three Shakespeare words" contains four words Shakespeare used, and also his name. If you mean words which are peculiar to Shakespeare, that is a bit more tricky. I like the word "superflux" from King Lear (it means the part left over, the extra). "Distemperature" is another (used in A Midsummer Night's Dream), which means disturbance…
Do you mean, "Did Shakespeare perform in a play called The Playhouse?" No, there was no such play, and the film of that name starred Buster Keaton and was made in 1921, long after Shakespeare's death. Do you mean, "Did Shakespeare perform in a playhouse?" Yes, understanding the word "playhouse" to mean a theatre where stage plays were performed. He was an actor and appeared in many playhouses.