1. Yes
2. No. You would be wise to get your
UK passport, but it would not be required.
3. Yes, right now,they are UK citizens
by descent. They would need to apply for their passport to live and work in Britain. Alternatively they could get an ancestry visa instead, go live in UK, and then apply for UK citizenship
otherwise than by descent after living there a few years. The advantage of the second method is that their UK citizenship woul become transmissable to their foreign-born children, which would not be the case if they took thir UK citizenship right now through you. The disadvantage is that they would not be full UK citizens until that time.
4. UK citizenship allows one to live and work anywhere in the EU, without additional visas