The only time she allows me to hold her (which isn't really 'allowing' it's more 'has no choice') is when I scoop her up while she is sleeping. She doesn't really bite, but will sort of put her teeth against my skin in a kind of 'I can bite you if I so desire' type way. After I put her back down she immediately starts washing herself like 'eww, people germs' and I, similarly, go wash my hands :)
Guess we both have slight germaphobic tendencies.
She is quite curious and smart, though (other than the digging thing, but I did read that her species does like to tunnel. Actually, I think she is more mouse than hamster because she doesn't stuff her jowls with food. Not at all.)
Her habitat (I prefer calling it that than 'cage') has a drop down door with latch. It is the access with which to feed and grab her while sleeping :)
The other evening I heard a lot of rattling going on and there she was, clinging to the inside of the door, biting at the latch. Clever girl. I think she was trying to open it. Maybe she is an escapee from NIMH!