This is just my take, and FWIW applies to my parents as well as my in-laws.
Tending to children's needs during family gatherings (e.g. preparing their meal and making sure they are fed, putting them to sleep, putting sunscreen on them, changing baby's nappies). Is it OK for GP to take on some of these tasks or can that be perceived as interferring?
All fine, but check with me first. So saying, "Would you like me to sort out X's food for them?" is great. Just going ahead and doing it feels like you don't think I will!
Holding babies and physical affection with children. How much is too much?
Too much is when the child is fed up. So unlimited cuddles but if my baby is crying because they want milk or my toddler is squirming and saying, "Down," then you need to stop.
Calling to "request" family time - suggesting they visit for holidays, weekends. Organising meetings so that they also visit extended family. Visiting to take children to the park/ cinema.
Suggestions are great but you have to be able to take a refusal gracefully. Sometimes they might have other plans, or just be tired. I do find my MIL doesn't realise how tiring DH finds big family gatherings, for instance, and will really push them when he doesn't feel he can face one.
Calling/ texting to check how doctor appointments/school functions/other events went
I'd rather a text, and let me take my time answering. In particula if something hasn't gone well, I'd like some time to process that myself before I have to go round telling everyone. (Applies particularly to doctors' appointments.)
Hope that's helpful, though as I say it's just how I feel about things.