Dog chews like Whimzees are definitely a good idea. Most hamsters don't like chewing wood and find them much tastier. It should be ok to leave in the cage, but you do get some crazy hamsters (like mine) who will pouch the whole thing and eat it in one night. What I do is only give it to Jasper to chew during play time or taming time (I'm still working on getting him comfortable to be turned on his back for health checks which is going well) and then take it away when I think he's had too much. This way I can monitor how much he's eating of it. Otherwise it will be hidden away in a hoard, which I don't want to disturb.
A cardboard box or two and some kitchen roll tubes might be nice? Or possibly adding a level as a feeding and drinking area?
I love your cage by the way, especially the wheel. It's such a pretty colour!
Good luck with your new hamster
!