On Game of Thrones if a girl marries a bastard, does she keep her maiden name? Or what?

Hardly anyone except the nobility (lords and occasionally knights decended from lords) use last names, and since a bastard can't inherit it's not a likely situation that he's important enough to need one. Most bastards aren't raised like Jon was unless their mothers are nobility aswell, otherwise they're left with their commoner mothers and if they're not ignored completely they'll end up at apprentices to craftsmen or something similar. Even the children of a bastard raised as nobility like Jon, Edric Storm, or the Sandsnakes are unlikely to end up as more than knights so they have no need or last names either.

As for the wives of bastards, if a noble girl was to marry a bastard she'd probably keep her name because that seems to be the custom anyways. Catelyn is called "Lady Stark" by northerners but in the south she's "Catelyn Tully" usually, Cersei and Margaery keep their names even when marrying kings, and the Targeryens would probably have been outraged if anyone without Targeryen blood tried to use their name. Infact the only person I can think of who is called by her husbands name more than her maiden name is Lysa Arryn.

The only bastards we know of who have, or tried to become lords or heirs (which is the only time when last names are really needed) are Ramsey and Daemon Blackfyre both of whom have been legitimized by the king, which means they could use their father's names. Ramsey does use his father's name and Daemon chose his own name.

Well one thing is that only nobles (and their bastards) in Westeros have surnames. The common folk don't seem to have them. The only example I can think of of an apparently common Westerosi with a surname is Masha Heddle.

It's also not clear if women always change their surname when they get married. We've got examples showing both trends. Catelyn, for example, is known as Catelyn Stark. But there are other women, such as Cersei, who continue to be known by their maiden name.

That's a really good question. I'm pretty sure she would. It's not like she would want to have the last name snow/stone/rivers etc.