Abstracting from Japanese games, which I don't play (I just don't like them), then the love/hate relationship has a right to exist, why not. Such an option can even make the game's plot more interesting, more suspenseful.
Yeah I can't vote yes or no. It depends how they are written. In some games the tsundere is a emotionally wounded person that just needs to be listened to or given attention that has been absent in her life. But sometimes they're just mean and vindictive for fun and you have to "tame" them. Yeah no thanks, if the game doesn't let me disengage and just interact with them minimally then I consider dropping it.
depends how extreme the tsudere is, if its more like highschool crush levels thats ok, but i've seen some variants that come off as full on hate, until all of a sudden theres love and that type i hate