This morning was particularly brutal, he started on me at quarter of 4 and would go away and come back about every 20 minutes until I finally got up and put him out at 5:20. I think I might close the bedroom door tonight and just let him scratch it. At this point, I'd rather sand down the door and restain it then have to be up at 4 am again.
He is an only kitten and I watch him outside, he has so much fun, he runs back and forth and up and down the trees, digs in the yard for moles and just generally is living the life

It is very quiet in my home because #1 we aren't home most of the day and #2 its just myself and DD. On days when it is raining and he CHOOSES to stay inside, he hides under DD's bed for the day. That seems to be the other problem, he has his nights and days mixed up, sleeps all day, plays all night and no matter what I've done to keep him awake early evening he just runs and hides.
Wow, I've always had dogs and was told that having a cat would be much easier. Not really sure just how true that is. My dogs trained easily, were affectionate all the time, and loyal. The kitten has a mind of his own
Dogs are easier to train because they want to PLEASE their master/Alpha dog. Cats, more or less, like the please themselves. So it makes them harder to train, but, as I've already proven, with a highly trained cat, they are trainable.
Training a cat is more about
rewards & punishments. They like being rewarded. They try to avoid the punishments. You need to get your cat to understand that you expect certain things from him. He does certain things, and he will get rewarded.

He does other behaviors and he gets punished.
He is at the perfect age to start learning these things. I started training my cat to sit at about 9 -10 months old. I got the directions off the web. I did it with treats first. Then I started using her food bowl and would not put it down until she did the behavior. Like Pavlov's dog and ringing a bell, my cat pretty much now sits on command without even thinking or realizing she's doing it. I constantly say SIT that much.
After about a couple weeks of only doing the SIT command, I then added LAY. (I had to push her from a sitting position, grab her paws to lay down, then
immediately reward her.) She would get confused at first because she knew I wanted something, but couldn't connect the word LAY to the action. But,
she did understand: I wanted a certain behavior from her and if she figured it out and did it, she would get rewarded (get to eat or a treat, or petting and affection.
)
Once ANY cat has learned that
concept, they can be trained to do
many things. Then, it just takes time, a lot of time and patience and repetition.
If they can figure out what they can or need to do to continue being rewarded, they will do it. Your cat has already figured out that crying or scratching at the door at 4 am gets him outside. He's got YOU trained.
You have to stop rewarding that behavior, and if necessary punish the bad behavior. Other people have said squirt guns, treats. I have found
several balled up socks thrown
in the direction of and above the cat and saying a firm No!, works at 4 am. NOTE: I said "in the direction of the cat." I did NOT say, HIT the cat. (This is the DIS, someone surely is going to mis-read and say I said to hit the cat.

) The balls of socks hit the WALL or the DOOR. They make a huge thumping noise if you have a hollow core door, as they hit the door. Both, having a projectile object coming
near him, (he will duck & run,)

and hearing you say NO!, as well as the noise of the thump are good reinforcements, that he does NOT want to continue that behavior. Have several balled up socks, because you WILL have to repeat this behavior quite a few times, when he realizes the balls have stopped, you've gone back to sleep, and he will try crying again.
Pick up the balls of socks each nite and replace on the nightstand. He will probably try this for several more mornings

, until he realizes that crying by the door will not get him out; it will only cause socks to come flying at him.
Then, he may move to knocking all the balls of socks off the nightstand
before crying at the door.

But, that is another story.
Eventually, you will get to a point when you just have to say No! without throwing the balls of socks or squirting the water gun.
It also sounds like there really isn't much stimulation for him inside, other than to come in and eat. No wonder he wants to be outside all the time. That's natural. I know you don't have a lot of time, but you & DD have to make an effort to play with him
indoors more. Especially right before bed.
I used to use a cat toy on a string. My kitty would chase the toy for a full 20 minutes and be really tuckered out. Then we'd
both go to sleep, and she'd sleep through the night. She still does that now, without all the playing. She has learned my bedtime ritual and knows when it's time for bed.
He will sit and complain by the back door until finally I just can't take it and let him out. If I try to go out and catch him he runs and acts like it is a game of chase. (When I turn to go back he will run after me...)
My cat & I play Hide & Stalk like this all the time. It's part of my favorite times with her. She always makes me laugh when we play like this. She LOVES to RUN

and "pretends" to hide, knowing full well that I know where she is. When I get up close, she turns around and chases me, and I hide. Although, she really does have to look for me.

It hones her stalking instinct. I jump out at her, and she's off running away again, like I scared the dickens out of her. Only she really isn't scared. She is hilarious!

If I had a bigger place, I'd be chasing her all over the house and she'd love it. It helps her to release excess energy, and again tire her out.