You need to become thane to get her, so yes you'll have to buy the house, otherwise you can't become thane to get her.
Yep she sure will. And if you want that extra bit of privacy *wink wink nudge nudge* you can make her a steward as well. Just remember you need a main hall first
Marrying a housecarl is the worst choice you could make. Their dialogue doesn't even change.
25.000 you say? Don't forget to furnish the house, so in the end it's gonna be about 30.000 in total.
Don't know why people like her. She's so bland.
She'd probably appreciate moving out of Proudspire, given she is forced to sleep on a bed roll supplied by Elisef the Cheap if she stays there. Every other housecarl is given a proper bed(or it is possible to make one in the Hearthfire houses) and some even have a separate room that gets furnished, Elisef the Unfair gives neither, just a bed roll in Proudspire's basemant.
Because dialogue doesn't change for the most part, it can get quite awkward when your wife starts addressing you as milord or, my thane...