Cagewise - for a dwarf hamster you want to be looking at an enclosure that is at least 70cm by 40cm in dimension (length and width) and no taller than about 38cm. For set up - one shelf across the cage is good, a wheel, house, plenty of substrate and floor toys - cardboard tubes, hidey places etc. And food and water. So starting with an empty cage can be good.
Are you finding it ok to find substrate where you are? Paper based ones are good if you can find them. Any wood shavings need to be kiln dried and dust extracted (if it doesn't say that on the bag, don't get them). Non kiln dried pine can be harmful as it gives off chemicals that are toxic to hamsters. Hardwoods are fine. Cedar is a big no no (so you need to know what kind of wood the shavings are made from).
Chinchilla bathing sand is the safest for hamsters, or if you can get childrens play sand you can bake it in the oven to sterilise it. He might enjoy a sand bath. Just any old receptacle with some sand in. And will get into good cleaning habits himself then - they often bathe and toilet in the sand bath - but toilet in a separate bit of it usually. They nearly always poop in it, but may pee anywhere! Luckily they are such small amounts that if you can't find where it is, don't worry about it - but just spot clean any wet patches of substrate once a week maybe.
If his current cage is too small then the best thing you can do meanwhile is enrich it - add more substrate and a couple of floor toys - cardboard is your friend! Have a look at erin's hamsters videos on making toilet roll toys!
10 Toilet Roll Toys For Small Pets - YouTube
I think your hammy needs some positive distraction to help come out of himself and to balance out the negative distraction he's had with the cage clean etc. A food parcel in a toilet roll tube is one - they have to get into the toilet roll parcel to get at the food (not his main food - something extra).
When moving cages the way to do it is don't clean anything at all in the old cage - resist! Doesn't matter if it smells for two weeks but it's really important to just move everything over from the old cage to the new without cleaning anything. And try and set it up in the same layout as before. To keep everything smelling and feeling familiar. So for example if the house was back right and the wheel back left - keep that same layout. As much as possible (as you'll maybe have a new shelf or something in a new cage).
The new cage will be bigger so will take more substrate. So add some new substrate but move all the old substrate across and spread it on top - so it smells familar.
The way I do it is to give them an adjustment day - and this really helps. It can mean the difference between a hamster settling in in a few days or being severely stressed for two weeks (ie cold turkey).
So an adjustment day is. You set up the new cage with substrate and one or two toys (maybe new things) - and put one familiar toy in from his old cage. Then move the hamster in there for a play session, to explore it. If he freaks out and wants to come out again then put him back in his old cage and try again later in the day. He may just enjoy it and explore first time, but second time he will be fine.
So let him have some time in there exploring (he will sniff and check everything out and be scent marking it a bit. He may look happy and perky! Good sign.
But then put him back in his old cage for the night.
Next day move everything over. He's had a bit of familiarisation in the new cage so will settle much quicker.
When moving everything over you need to find somewhere to put the hamster. If you have a pet carrier you could pop him in there. If not then a play box maybe (high sided so he can't get out) with a bit of substrate and a tube and some food scattered.
This is another psychological thing, but what I find really helps the process is - put the hamster in another room in the box/pet carrier. So they can't hear/smell/see what you're doing! They KNOW when you're messing with their cage and get quite stressed. In another room, where it's quiet he can't smell and hear the upheaval.
But only do that if you know he's in something secure. Or if there's someone else in the room to keep an eye on him in case he does try to escape.
So with hamster out of the way, move all the old substrate out of the old cage into the new, on top of the new substrate - just press it down a bit if it ends up being too much,
Move over house, wheel, toys etc, get everything set up safely, add any new items. Keep his old nest and move that across too, inside his house and most of his hoard as well. Even if they smell! And put his hoard back in the same place.
Don't wash or clean anything - food bowl, water bottle etc. They'll be fine for a couple of weeks.
Technically they need another two weeks settling in period after a cage move, but if you do everything like that it may only need a few days and the hamster is a lot less stressed.
He'll probably go awol for a few days and hide away but then start coming out more.