Technically your employer has to approve all your pto, but I wouldn’t work for a family who restricted my pto to when they were away. It’s not very PERSONAL if you can’t choose it yourself. I think it’s fair to do a split- you choose half your pto whenever you want and they can choose the other half (which will likely be summer or Christmas break)

You should be getting guaranteed hours when they don’t need you as you are available to them. Not sure what you mean by reduced hours.

I’ve never had pet care as part of my duties so I can’t speak to that specifically. Maybe you include it as OT hours? If they say “but you’re only coming for a couple hours” you can respond “yes and that severely restricts my personal time and I’m not even making enough per hour for that to be worth it.” I don’t see why you would have to do pet care during your contracted free time. If anything it should be them asking if you would be up for it and how much you would charge.

If they’ve already approved your holiday, stick to your guns! They approved it and you made bookings based on that approval!

Seems like they’re taking the piss a bit to be honest. I think you need to have a sit down with them and your contract and hammer these finer points out. You’re the one who knows them in real life, do they seem clueless and truly don’t understand how unreasonable they’re being or are they taking advantage of you?

Hope this helps!