Hi Zoe, I feel for you, it sounds like an awful situation.
I used to work in a professional kitchen (for about 15 years) and I agree, i would never be able to accept some of the hygiene issues you mentioned (I am still cringing about the cloth used to wipe the dogs' bowls!) I don't think you should have to accept it, either, not least because if there is ever a food poisoning outbreak, it could well be blamed on you as having the responsiblity - even if in practice you can't exercise it.
I think Jane has hit the nail on the head - they can't deal with her as it's family, so they've copped out by shifting the problem onto you. I also think if you don't take a stand now, it'll only get harder and harder to do so. I'd suggest fighting the easiest battles to win - the hygiene ones re dogs and bins - you can truthfully state it's a hygiene risk and that you can't professionally accept it. Throw ANSES into the conversation (www.anses.fr). I'd also recommend documenting and/or making notes on every conversation relating to health and hygiene, as a back-up and again to protect yourself.
Realistically, if they won't change, how much do you want/need the job? I'm just thinking that for someone with your skills and knowledge of the food/accommodation industry, the start of the summer season is such a good time for finding another job -even if it's a bit of a stop-gap to tide you over for the summer while you look around for the next challenge.
That's my thoughts anyway - let us know what you decide and how you get on!
All the best, Diane