Hello and welcome to HC!
Firstly, a month isn't that long. While he may be more content to do his own thing, I'd still say he's in the settling in period
Do you have a free range area for him?
In terms of transporting. I use a large glass jug that they walk into while tipped on its side and if needed a book can be placed on top. That should solve that issue.
In terms of awake hours. I've had some really late risers, 3am types! What I do is usually pop in the room around 9pm and make a bit of noise, sometimes a curious ham will pop out and if they are still awake after a few minutes, had a drink etc,ill get them out for a short time, 5/10min, then pop them back. I use their "sleepy time" to handle them also.Pop a blanket on my lap and let them run over me.
I find once they feel comfortable in their home, they'll start popping out once you are making noise in the room.
I don't advise lifting lids of houses either.
In terms of spot cleans. Once the jug is mastered, just pop him in the bin cage or free range to do it then.
I'd just relax and try to do taming/bonding outside the cage so he can get used to you. Some hamsters act differently inside their cage too so I work on our bond outside