I have a power bank that claims 300000 Mah capacity - it's good for fully charging a mobile and/or tablet a couple of times with juice to spare, This does not have the capacity marked and it sat in my carry-on laptop/personal bag on weekly trips to Europe (mainly on BA) for over 3 years with no bother.
1/ yes
2/ I never bothered. The (installed) laptop battery I was also carrying was probably at least as great a risk.
3/ YMMV, but be aware you will have to keep it with you if a bag gets pulled at the gate to go in the hold.