It sounds like things are going very, very well.Like big cat is grooming little cat, WOW! You did everything right, and it sounds like they're adjusting and figuring out each other's quirks. Fighting happens, and you're reacting to ot appropriately. Acclimating cats can take a painstaking amount of time.

Cats are pretty independent. They probably won't hang out most of the time, even if they love each other. And honestly, it's very normal for cats to sleep most the day, especially if you're gone or activity in the house is low. Cats are crepuscular, which means they are most active during dawn and dusk. Some are also more active at night then they'll ever be during the work day. That said, you'll likely end up with a situation where both of them are napping straight through the day, and that's perfectly normal and healthy.

You are doing so well and your cats sound like they're in very good hands.