The diet is basicly the same in the wild, domestic rabbits survive in the wild pretty good if predators don't get them. Grass, weeds and whatever they find is all they need.
There are differences though, mostly for convinience of the owners. Hay doesn't exist in the wild, but it it is needed here to feed rabbits (and other domestic animals) through winter, and it has some additional merits besids being good food, easy storage and all year availability, for example it's safer than fresh forage considering some diseases in contaminated regions.
Pellets and veggies are convienient pet and livestock feed too, and since rabbits can thrive on a variety of diets "natural" doesn't necessarily mean healthier than others as long as some thought is put into it.
It depends on what they get used to.
What, and how much you feed kits depends on your goals, if getting them to butchering weight asap a free fed protein rich diet might get best results (after starting with whatever the doe eats), if fastest growth isn't your prime concern it doesn't matter that much. I wouldn't aim for fastest growth with pet rabbits.
Most rabbits don't like being picked up, there can be a selective bred in tendency to accept it better or at least to get used to it , but mostly it's up to the individual character. That a "wild" rabbit was ok with it is remarkable, what continent are you living on?