need help with installing window trim
What is the wooden frame around my window?
That's the window "jamb".

Should the wallboard terminate at the rough opening?
Yes. You don't normally extend your wallboard all the way to the jamb so you have room to insulate, etc. 

Will I need to extend the jamb to be flush with the wallboard?
Yes, extension jambs are often needed with retrofit construction. Ideally your jamb will be a uniform distance behind the wallboard all the way around. If it varies by more than 1/8" I'd address that. 

Do I need to use thicker trim to account for the thickness of the tile?
This is entirely a question of design preference. Normally, tile isn't as thick as the standard trim and therefore whatever is installed elsewhere in the home works fine.

Should I fill the gap around the window with some kind of sealant or insulation?
Yes. Modern practice is expanding foam of the "low expansion" variety. You don't want to bow and bind your window.

What type of wood trim is recommended?
Again, design preference. It can be stained or painted, hardwood or softwood, or whatever you like. It's not exposed to weather, but consider whether it'll be subjected to moisture and choose accordingly.