Dwarf hamsters require a wheel that's around 20cm. Add in the stand, and there wont be room for it and the roughly 15cm deep substrate that hamsters prefer. So, yes, try to make it taller if you can. =)
Mesh wire should be fine, assuming you mean metal mesh. Just make sure the gaps are no bigger than 1cm, or a particularly stubborn hamster might try to squeeze through.
Re: Building enclosure
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