I am also not a fan of hamster balls for out of cage time but it's very useful to have one for safely transporting them from one room to another (ie carrying them in the ball) especially when they're not tame. In that case the size doesn't matter and the 7" would be fine but too small for running in. The problem with the 12" rat balls is the slits are too large and they can get injured by getting a toe caught while running. Of the 12" rat Balls the Savic one is the only one that has smaller slits and not sure if that is available in the US. I am still a bit iffy about that one though.
I would put a litter tray where she pees now - they will only use it if you put it in the place they have chosen. It is possible she may move her pee place if you put the wheel there, but also very likely she will just pee under the wheel or in it! Is there another place you can put the wheel? I think it would fit at one end (ie against the short side facing the other short side) and still leave room in the corners for a pee place - assuming she is using a corner, which they often do.
The bendy bridges stay up on their own because you bend them curvy and that gives them their own support - so like an s shape bend. You'll probably need two overlapping as the s bend makes them shorter effectively. You don't want the really wide ones or the fence is so high it's an issue getting back over it! So the 10cm wide ones should be fine - you can still have substrate a bit deeper than 10cm by sloping the substrate up away from the fences towards the other end. I think for a Syrian I would just use lower substrate on the other side of the fence and have sand in the litter tray. Syrians don't really need a sand bath so having sand in a litter tray is good. Enough for them to mess about with and pee in
It needs to be Chinchilla bathing sand (not "dust").
Mineral chews aren't really necessary and I have yet to meet a hamster who enjoys the wood sticks so a treat stick with some tasty food stuck on the outside should go down well - and help keep the teeth trim. Ideally sugar free though. Also whimzee mini toothbrush dog chews are very popular with hamsters - also come in different shapes like alligators. I like the toothbrush ones because you can stick sunflower or pumpin seeds in the bristles sometimes as an extra treat and they enjoy wrestling the seeds out!
These are the whimzee chews - vegetable based. They only seem to sell enormous boxes on Amazon when you can get them in packs of four or singly so you might want to look in a pet store for them.
https://www.amazon.com/s?k=Whimzee&ref=nb_sb_noss
I get these Rosewood naturals ones although they cost about $11 for two with shipping in the US presumably because imported. They're sugar free.
https://www.amazon.com/Rosewood-Bore...ateway&sr=8-26
These are good too but seem a bit expensive (maybe imported too)
https://www.amazon.com/Vitakraft-Ham...ateway&sr=8-10
When looking for cage items and hamster toys, here are a few pointers. Any round entrance holes need to be at least 6cm diameter. Any wood items need checking for splinters (since having a hamster I now keep a sheet of sandpaper handy!). With wood items also check for protruding nails.
Actually I have always really liked this little extra hide! Not big enough as a main house but they like other hidey places to visit and sit in and it has a flat top so they can climb on it too. It's only sold in the US I think. The entrance will be fine as it's open at the bottom, so if on the substrate they can easily get in and out by burrowing down a bit if necessary. I have a similar little hideout with a flat top in my cage and always put an "extra" or treat on it, so our hammy knows where to go for food, where to go for his veg and where to go for his extra - keeps them using the whole cage.
I put his bit of daily veg in a different place to his hamster mix and his "extra" (varies between half a walnut, half a pecan nut, a small bit of apple (no pips) or a blueberry eg somewhere else again.
https://www.amazon.com/Kaytee-Natura...y&sr=8-83&th=1
Scattering a bit of food as well as putting it in the food bowl means they enjoy foraging for it and stimulates them. Sometimes I'll scatter a few pumpkin seeds instead - they adore those.
If you didn't know already, it's good to supplement their main hamster mix with a bit of fresh veg daily and they really love fresh food. At first you just want to give it every 3 day so their stomachs adjust to it. After a week you can go to every other day then after another week, to every day.
Just a very small piece though - eg a tiny broccoli floret the size of your thumbnail or 1cm cubed size piece of carrot etc (doesn't have to be square! - that's just for size).
I tend to alternate between what I usually have in the fridge that keeps a long time, which is broccoli, carrot, baby corn, cucumber - they keep much longer in the fridge. And fresh spinach if I'm treating them as a bag only lasts a couple of days when open. There are many other safe veg as well and some unsafe, but the ones above are all safe.
An occasional bit of apple (apple pips are poisonous though so it needs to be cored) similar size no skin, or a blueberry or bit of banana are ok as well.
What hamster mix are you using? Syrians aren't prone to diabetes like dwarf hamsters but it's still good to use a sugar free mix as they can get toothache and dental decay like humans when they're older!