It seems like rule in the Dominion follows the simple rule: Don't get too powerful, don't cause trouble, and you never see us. Bother us in any way, Jem'Hadar make you go boom. It's clear in several episodes that there is free commerce within the Dominion, and the Dominion probably encourages things like the Tosk hunt like the Roman colliseum. Social freedoms and economic freedoms, but no political freedoms and constant fear of seeing the Jem'Hadar on your doorstep.
That sounds about right, from what little evidence we have on the life of Dominion subjects.
The Dominion is probably way too large for them to rule the whole thing with an iron fist, so they let most species go about their own business with little interference, and bring the Jem'Hadar down hard on anyone who causes trouble.
As for how they become Dominion members... it's probably as simple as a delegate of Vorta going to a planet (leaving a huge number of Jem'Hadar in orbit) and politely informing that planet's government that they are in the Dominion now.