My DD is back home post uni, working and saving towards the post grad course she is taking next year. We don’t charge her rent - she buys quite a lot of her own specialised food, cooks for us sometimes and does the occasional big shop to contribute to general supplies. This is what we are happy with - we don’t need a financial contribution from her and this is her family home.
She has some set areas of cleaning responsibility , comes and goes as she pleases - texts to say if she won’t be coming home so that I can stop half listening for the door click, has her bf to stay when we are away and occasionally when we are here, and has had a couple of small and unriotous, dinner parties by prior agreement, is currently redecorating part of her bedroom.
We treat her with respect due to the adult that she is but she isn’t on equal terms with us because it’s not her house,its not her financial responsibility and we are her parents not her flatmates.
The question of the appliance purchase is an unusual one in the case of the OP . I would say that it isn’t at all a common arrangement but if the family income is very low and buying a necessary thing like a fridge is going to be a struggle then it would be reasonable for her to chip in. The 1/3 split of utilities and rent also is very unusual rather than a contribution but, again, if there is such a low family income may be justification for more than token board money.
It is a tricky one - I can see why she is not happy with the flatmate type cost split but without the equality of a flatmate . The father needs to rethink his attitude a bit and either treat her less like a complete child or take less of a contribution.
The pet thing is a bit of a red herring - you’d have to have landlord agreement in a rental or agreement of flatmates in a flat share so, no, I don’t think that is a solo decision.