Without further context it's hard to understand why OP is so resentful of her father. Breaking down the gist of things doesn't help.
She keeps saying "we don't see eye to eye" but from the examples given, there's not a justified reason.
He has a favourite mug and doesn't want her to use it. I can't see that as a major problem.
He passed comment on the appearance of her car. Okay, that's not really necessary but other than being annoying, not really noteworthy.
He bought something he thought she'd like and she rejected it. I don't see why. That's just cutting off your nose to spite your face.
She told him they don't have to talk. That really seems to say more about her.
Whilst the parents were separated (not divorced, as far as I can see) her mother had a relationship with someone else. I don't see any indications that OP is upset by this.
It seems that OP perhaps feels that her father is dismissive and rude, and possibly doesn't treat her mother how she'd like? There seems to be resentment about them getting back together. But the key thing here is that this is what her mother wants.
It seems as though it would be better if OP found somewhere else to live whilst work is being done on her house.