[Tazius]

I have suggestions for the first two points

Firstly, don't worry about the wheel. I think we got our hamsters at the same time and Faolan is exactly the same (smudgey poop included). Someone suggested to us it's likely because a) hamsters love to run, and b) new hamster have never had a wheel before so will be absolutely enamoured with it. Faolan is alwaysssss on his wheel, and had pooped and peed all over his too... In terms of poop, is all of her poop greenish or is it just the poop in the wheel?

Secondly, definitely invest in a bottle as soon as you can! You can pick them up real cheap an use them regardless of if you have a wired cage or a bin cage