He is doing this because he doesn’t want to be closed in a cage! No matter how large your cage, I’m betting it’s not large enough for all the running about he gets up to when you sleep. My bunnies run for hours on end during the night, and my two are free-range so always have opportunity to exercise!
I understand your concerns about letting him out to roam free, there are dangers to rabbits in every day objects, especially for young ones like yours. However, if you properly bunny-proof your room or approved area for him then there are huge benefits to letting him explore. Firstly, he will be mentally stimulated which reduces things we see as destructive behaviour. He will become familiar with different sounds, sights, people etc so is less likely to develop into a nervous bunny and lastly, he will get ample exercise which will aid his health and wellbeing. I cannot highlight enough how content my bunnies are being free! To stop him chewing on things that aren’t for him (because trust me, he will try!) just distract him with an appropriate chew toy and praise his good behaviour with petting and small treat. Soon he will realise what is for him and what isn’t. My rabbits went through a phase of chewing everything even after the training, and I mean everything: wooden table legs; chair legs; doors; skirting boards; books… I then realised I was not giving them enough variety. So remember to change the toys regularly to keep him interested also. But if you want a house bunny, you will have to learn to deal with a little bit of nibbling here and there!
Right now, I would suggest only letting him out free when you are there to supervise and make sure he doesn’t get into any trouble.
For times you are not there, I would set up a puppy pen around his cage and leave his door open. It means at least he has more space than just his cage area but also provides him with a safe place when needed. Puppy pens can be bought pretty cheaply (£15-£30) on the net but always read the description properly and make sure you get one with small spacing inbetween the bars. There have been some horror stories of bunnies getting their heads caught inbetween the bars and dying because the spacing was too large!
Other than these things, you need to bare in mind that he will start to become more hormonal as he matures and will have to be neutered to help stop this and the relating behaviour. Unfortunately your beautiful bundle of fluff can start to act like a little monster during bunny teens but do not give up on him, it is only temporary and after he has been neutered he will return to being a little delight. I say this from experience 