Ok he's a russian dwarf hamster
Not a chinese - so doesn't need to climb. Unfortunately Russian Dwarf hamsters can have a bit of a reputation for being bitey. It varies, and yes your daughter is doing a great job of taming him. If he still bites then she could wear gloves (something like suede or gardening gloves) and hopefully when he is more used to it the biting will stop.
@Doodleham on here does that.
The Hazel Hamster you linked is fine. However it is better for Syrians than dwarf hamsters. It has some large-ish pieces which are good for Syrians. You can technically give it to dwarf hamsters as well, but ideally a dwarf specific mix is better for dwarf hamsters. But there aren't any commercial dwarf specific mixes in the US without sending off for some from Germany I believe. The Higgins Vita Garden would have smaller pieces so would be better for dwarf hamsters, and also suitable for Syrians, but if it's easier to get the Hazel Hamster then they could all have it. And what you could do for the dwarf hamsters, is give them some extras - they like small seeds - so you could get some linseeds and hemp seeds and mix them and put a pinch of those out daily on a separate dish. They are also healthy seeds for immunity and fur condition. They are fine for Syrians as well. I tend to give the linseeds and hemp seeds to older syrians or ones with health conditions, but your Syrian could have them as well just for general health, but not essential.
Socks Mum is right that string is bad for hamsters to chew, but it looks like the nice hanging chew/boredom breaker you have is sisal string, and that is ok. So I wouldn't remove it just yet. Is it a kind of rough hairy string?
Of the other items you linked.
The first one - no. Those hamster toys have chains and aren't very safe or anything a hamster would actually use
Second one - the platforms - as with all wood hamster items you need to know what would it is. I think it just says "hamster safe wood" - that isn't good enough as "wood" could be anything. It needs to say pine wood or birch wood or whatever. They might be ok but I wouldn't rely on it. Also with platforms you need to check the leg length - that's the tricky bit, finding one the right height. To answer the question, platforms are perfectly safe - as long as they are not too high. No more than 2" above the substrate level. Even an inch above substrate level is ok because there is still a gap for them to burrow under or you could lower the bedding a bit under the platform
Third one - bendy stick bridges. These can be useful as ramps but can have little gaps in where hamsters feet can get trapped so people stuff moss or tissue paper in any gaps. So that is a possibility, but there are some smooth ones that are safer and don't have gaps. Linked below - most of the Niteangel items are safe.
This is a nice one for dwarf hamsters - its both a ramp and a hide - but it's too small for Syrians
This is also a useful little ramp for either dwarf or syrian hamsters and some people use it for access in and out of a sand bath as it's steeper on one side and longer and shallower on the other. By the way do they have sand baths? Hamsters do benefit from a sand bath in the cage all the time, and also sometimes use it as a toilet and then you can just use a small sieve and sift out the wet patch!
This is the smooth bendy stick bridge without gaps.
However Niteangel stuff can get quite expensive to buy - not all of these are essential.
A page of Niteangel hamster safe items
The sprays are fine. I would probably go with the Bucatstate selection as it has flax in and dwarf hammies seem to love nibbling the flax!
Forget about the climbing
That was just for a Chinese. Syrian and dwarf hamsters are ground dwellers and yes they can fall from heights and hurt themselves. But a low level platform is fine. And no risk there even for an older hamster.
