Hi! I've been browsing this subreddit for a while, and created a username because I've finally come across a question I can answer!

I can only speak to the western tradition - my education was sorely lacking when it came to Asian theatre.

The answer to your question depends on which point in history you're looking at. I can speak to a couple of time periods.

Ancient Greece - Most theatre at this time took place within the context of Dionysian festivals (celebrating the god Dionysus). Financing came from the state. After 449 BCE, the cast was selected by the state - most likely to reduce one playwright/director having an unfair advantage over the others. Actors were also paid by the state, however, lead actors - and only lead actors - could compete for a best acting prize, which could be awarded to a play that didn't win the festival. Note that these festivals would occur about four times a year, so most actors would most likely not have been an actor by profession. Based on historical writings, it seems that vocal ability was most prized. Greek actors wore masks, so facial expression was of no matter.

Middle Ages: Most of the theatre that occurred during the early middle ages was associated with the church. Liturgical dramas, etc. These would have been devotional in nature, and unpaid (think Christmas pageants). Even outside of the church, a lot of theatre was religious in nature, such as the plays that took place during the feast of Corpus Christi. These celebrations would recruit local guilds to take part in a processional, and in theatrical events. This would have been on a volunteer basis. According to Oscar Brockett's "A History of Theater," most actors taking part in these festivals would have been merchants or working class, although nobility and clergy occasionally participated.

Professional theatre in the Middle Ages and very early Modern period was mostly relegated to a few distinct forms: morality plays (public performances), interludes (private performances). Morality plays were first performed by amateurs, but were then taken over by professionals. Interludes were plays performed indoors for those who could pay for it - usually rulers. Interludes were often performed by minstrels. Doing a good job could mean being retained by a nobleman as part of his regular company of performers (this starts to happen by 1350). Nice work if you can get it! That said, professional performances were not nearly as common as amateur performances until the last half of the sixteenth century.

Oscar G. Brockett's "A History of the Theatre" is a great introductory textbook, and is where most of my information comes from - that, and what I remember from introductory theatre history five years ago. The Broadview Anthology of Drama is also a good resource - while it's a dramatic anthology, each play is introduced by some historical context.