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With the wheel I would just watch. If his back is bend while running then you should get a bigger one, otherwise it's fone. If you're worried with him flying out of the wheel, there are wheels where the side is closed and only some holes are in it so that the hamster can climb in but not so easily fall out :) like this, for example (it's bigger than you need, but it's just so I can illustrate what I mean xD) https://www.amazon.com/Wodent-Wheel-Senior-Tail-Shield/dp/B01MRWR7MQ
As for the sunflower head:not really. Any dried sunflower head should do. You should be able to get them from your local pet store in the aisle for birds. Oh and if your already there, grab some flax. Trust me, your hamster will go crazy for it. It's like cat mint, only that it's hamsters favourite snack xD
The sand you have sounds good. I just overlooked your sand bath 😅 Thank you for caring so much about your hamster. He will have a lovely home ❤

My recommendation is not to use a seed mix as they're only food source, and use a good quality pellet as the main food with possibly a small amount of a seed mix (no pellets) occasionally as a treat.

If your hamsters are being fussy it can be for a couple of reasons, a poor quality mix with too many cheap sweet fillers (poor quality corn for example) and poor quality pellets or over feeding.

In nature hamsters are omnivorous and foragers. Which means that they have an extremely vaired diet in the wild and eat many things they "stumble" across as they forage.

They also live in multi-chamber burrows with a number of lauders they store food in for later. They'll often choose better quality/more perishable/tastier food first and leave less desirable food for leaner times. So if they're being overfed you'll see the behaviour you're describing.

The major problem with this behaviour is the fact that they will most likely not get full nutrition eating in this manner and is why I go with a good quality pellet that's been formulated to meet all/most of their needs as my main food for my hammies.

I've had really good luck with this as the main food and then give approved treats daily as well. Most pellet mixes are low in the protein level that hamsters need. But that need varies with a high of 20-25% when growing to more like 15-17% when elderly. So giving additional protein off the list as a treat is a good idea especially when they're younger.

The other thing about using the "nicer" foods as treats as they work really well as a training tool. If you only give them pellets and the occasional seed mix, you can hand feed them the treats which gets them used to your presence and smell quicker in a positive way.

Edit: a word

Hamsters are nocturnal, so while you may think she's not using her wheel, she might be using it while you are asleep or away from home. I do hope you get a larger cage for her soon, as that cage is too small to accommodate an appropriately sized wheel. If she truly isn't using a wheel at all, the size of the wheel may be contributing to that.

Syrians should run on a 12 inch wheel. An economic choice is the comfort wheel from Kaytee:

https://smile.amazon.com/Kaytee-Comfort-Exercise-Wheel-Colors/dp/B0010OR008/ref=sr_1_1?crid=2HS8AQ2VOVHJB&keywords=12+inch+comfort+wheel&qid=1562230747&s=gateway&sprefix=12+inch+comfort+%2Caps%2C184&sr=8-1

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Although it will appear oversized, a 12 inch wheel will prevent your hammy from arching her back, which can cause issues as she ages. Regardless of which wheel you provide, make sure the wheel has a solid running surface, versus the wire wheels.

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You mentioned being on a tight budget, but if you can save up to afford it, this has been a great cage for my hammies and I think could really improve your hammy's quality of life:

https://smile.amazon.com/Prevue-Pet-Products-528-Universal/dp/B074FH47MQ/ref=sr_1_6?keywords=hamster+cage&qid=1562231054&s=gateway&sr=8-6

Awe, isn’t his little face the cutest! He looks so skeptical lol. I will second what another commenter posted about the wheel. Did you know that the current wheel you have can cause issues like a broken leg, bumble foot and a hunchback? I have linked a couple cheap wheels that would be more suitable for him. Ideally, he should have an 8 inch wheel, but you may not be able to fit it in the cage so 6.5 would work.

Kaytee Comfort Exercise Wheel, 8.5", Large, Colors Vary https://www.amazon.com/dp/B0002AS94C/ref=cm_sw_r_cp_api_i_LE4EDbG98Z896

Penn-Plax 8" Exercise Wheel - Perfect For Rats, Hamsters, Mice, Gerbils and Other Small Animals https://www.amazon.com/dp/B071RLP963/ref=cm_sw_r_cp_api_i_-E4EDbWRPX7P8

Kaytee Silent Spinner 10" Exercise Wheel, Large, Colors May Vary https://www.amazon.com/dp/B07DT7V35J/ref=cm_sw_r_cp_api_i_wF4EDb026AN7S

The main goal is to have something with a solid surface to protect their legs and feet.

For anyone in the US whose interested in this, it's sold in America too :)
https://www.amazon.com/dp/B07FQ411ZH/ref=cm_sw_r_cp_apa_i_nR17CbPZS5WN3

YES.

if you want to keep is simple, i found an amazing wire cage that works perfectly for my syrian. it has two separate higher stories.
this one: http://www.amazon.com/Prevue-Hendryx-SP2060R-Hamster-Bordeaux/dp/B005VEWFUS/ref=sr_1_10?ie=UTF8&qid=1371094703&sr=8-10&keywords=hamster+cage

it comes with two houses and a shitty wheel, so i replaced the wheel with a silent spinner and removed one of the houses and replaced it with toys and food (so he has to walk up stairs to get it.)
i also got him a small wooden maze that fit on the lower level, so it's like a jungle in there.
it's also SUPER easy to clean.
i highly recommend it.

now if you wanna get fancy, HABITRAIL CAGES!

They're usually half wire, half plastic. sometimes all plastic.
it's a bit of an expense, since no single habitrail cage is big enough for a syrian, but if you connect for example the loft and the twist, that makes a pretty decent space that you can also add more to. but keep the tubes to a minimum. try connecting big spaces. (i recommend the habitrail loft, habitrail twist, and habitrail ovo home. don't get habitraiol crystal: those are only appropriate for dwarfs.) the water bottles are kind of uncomfortable too, so i use an external one that came from some crappy Critter Trail cage (i don't like these and don't recommend either)

Thanks for the kind comments folks,

Everything was bought in the UK

Cage - https://www.amazon.co.uk/gp/product/B07FB6MTJ9/ref=ppx_yo_dt_b_asin_title_o00_s00?ie=UTF8&psc=1
Wheel - https://www.amazon.co.uk/Wooden-Exercise-Wheel-Large-hamsters/dp/B008H2VG0G/ref=sr_1_4?keywords=wooden+hamster+wheel&qid=1559158708&s=pet-supplies&sr=1-4


I imagine you could make the wheel tbh, does not look too hard.

He shouldn’t chew through it. For extra security, they sell lid clips for tank lids. Here’s a link to some on amazon. I guess they go by tank size, so these might just be an example for you.

https://www.amazon.com/dp/B016MDJH7A/ref=cm_sw_r_cp_api_i_ui.kDbX9D3CZR

I had that wheel too, it was very noisy and generally awful.
Trixie 61011 Rotating Ferris Wheel Hamsters, Plastic, ø28 cm, Assortment: Random Colors https://www.amazon.co.uk/dp/B00365P9NY/ref=cm_sw_r_cp_api_i_fWAxCb9DF3RY8
I highly recommend that one, it’s extremely quiet and perfect for syrians.

I actually got the cage used. But! This is it basically: https://www.amazon.com/gp/aw/d/B003TKRYKC/ref=mp_s_a_1_9?ie=UTF8&qid=1493687394&sr=8-9&keywords=guinea+pig+cages

I like how big it is so the hammy has room to roam.