My hamster wakes up at 12-1 in the night. She usually slows down at 3-4, and then she is active in the morning again. It changes a bit though, since she is young and she is finding her regime. It is normal for a hamster to wake at night. Some are nocturnal, some are crepuscular like mine (wake at dawn and dusk). The best time depends on you and your hamster. If I take Yasmin out at 1, when she is fresh, she is very very active. When I take her out at 4, she is sleepy already. Of course, I only take her out when she wants to, and she prefers to be out in hamster proof room most of the times. I can tell that she wants to go out by her jumping on my hand, or standing on two legs finding a way of how to get from her Detolf. Only this way the hamster will enjoy free roaming time. You should wake up your hamster only when medical emergency, because it is very stressful and harmful for your hamster. That is also why he does not want to come out when you wake him up - he just wants to sleep. Try to interact with him for example in the morning, when you know he is awake.
But back to proper time to play with a hamster. The best time also depends on you. If it is better for you to play with hamster in the morning before school, do it, it is important you have enough sleep as well. Also, hamster can be a bit trained - if you know, yor hamster is up lets say 1-6 in the morning, and you go fill your hamsters food at 5 every day, your hamster will learn this time and wait for you every day. So, to be honest, the best time is individual and it needs to suit you and your hamster regime.
Someone recommends a tiny tiny piece of veggies or fruit, but you should be careful with amounts and your hamster needs to be used to these. If overfed, hamster could get diarrhea. You can also feed your hamster just normal hamster mix, if he is not too bored with that. You can give a bit of pumpkin seed, but that is also fatty. A bit of egg or mealworm as a protein can be as well. If you type into google some type of food and amounts for your type of hamster, you can find how much per week or day max he can have, so that is very useful. It is said that you should give your hamster only that much fresh food that he can eat in time, so it is not rotting in his nest.
The best method is slowly try to put your hands below his belly. That way hamster wont be scared that you are a predator, trying to catch him from above. He should see and smell your hands upfront. It is possible that at the beginning, he will be uncomfortable and try to jump from your hands. That is ok. He will learn to trust you in time, it needs some training. However, when you want to take your hamster out, use some transfer box, normal cup or something, to prevent him from jumping from your hands on the ground. He will learn that cup or bow or travel box means play time and he will go there on his own, when he wants to go out play. You can put also a treat in there.
Your hamster can be scared to go out when you are awake, so he can prefer to just hide until no sounds are around. It is normal. He might be just cautious. If you want to clean his beddig, I suggest to lure him out of hist nest with a snack. I usually hold snack next to the opening of the house of my hamster. You do not want to open the house, vlbecause that will scare your hamster. I know this is hard, but you need to be patient.
You are amazing for stepping up and taking care of the hamster. We need more hamster owners like you.
I recommend you youtube channels like Victoria Raechel to search, or you can search through these older forums on hamster central to learn more. As I have syrian, I cannot unfortunately advice you on dwarf hamster diet. Good luck!