I remember now about the bar spacing. Never trust a hamster! You've been lucky. Are you sure he wasn't sneaking out at night and then sneaking back in again! Guinea pig cages usually have about 2cm bar spacing and definitely need meshing if it's for a hamster. Anything bigger than 1cm bar spacing they can get out of.
Usually a hamster's raison d'etre is to escape - no matter how large the cage. However I have accidentally left the door open all night before (to my horror) and my hamster didn't leave the cage at all. Some of them are happy in their cage especially when older, And I know he hadn't been out as the cage was high up. But they escape a LOT! Their most active time is at night when you're asleep.
It doesn't matter if the mesh is for chickens or birds - it just needs to be 1cm square mesh. However meshing a cage is not an easy thing to do - and it needs to be done so it's safe and no sharp edges (and it can hurt your fingers a lot).
Not something that could be done too quickly. The other option would be to make a large bin cage instead - but that would mean dealing with mesh at some point as well. And I think it can be hard to find large enough bins.
I definitely wouldn't get another hamster until there's a way to secure the cage bar spacing or an alternative kind of cage (if you found a different cage you could maybe sell the guinea pig cage?). But if you want to mesh the cage then once it's done it's always suitable for any hamster. The mesh needs to go on the inside. If you mesh the outside the hamster can still get through the bars and get trapped under the mesh and cut or hurt themselves (and be difficult to get them out again).
Most hardware stores should sell 1cm square mesh. And if you take the measurements you need, they could even cut it to size so you have all the right size pieces to attach. You attach it with cable ties. Then the fiddly bit is folding the edges under on the mesh so there aren't any sharp cut edges on the inside of the cage.
It is hard still thinking about your hamster at this time and the memories when it's so recent - in time you'll be able to remember the happy bits without feeling sad.
If the vet gave you some stuff to disinfect the cage that would be fine. I still do the soapy thing afterwards to make sure a little crevice hasn't been missed and it's all been washed, not just disinfected. That is the quick easy bit with a sponge. I do it in the bath so I can use the shower head to rinse it all off.
Yes for Syrians you need a 28cm diameter wheel (bigger is ok), rat sized hides and tunnels (pringles tubes make good tunnels
). And a large house. A shoe box house can be a good size - a cardboard one that is. Easy to make out of a shoebox. And big enough to fit a litter tray inside.