Where I live I think fridge/stove normally are left behind. You can bring them with you, but I think it's normally done only if they are extremely valuable (like a nerdy import or design stuff). A super pro chef might like to keep their super duper stove. Often stove and fridge are integrated with the counters and stuff, besides you don't always know if you can use them in your new house/flat anyway.
Yes, leftovers and fridge inventory in general will automatically move as they are per household and not per fridge, but I do empty them for realism. I might not take out tons of vegetables, but at least deal with leftovers and glasses of water etc. And - some times a left over dish will actually be the first meal in their new home, perhaps before everything is "packed out" :)
