Travel to London!

Hi Ladies,
I am travelling to London with my 10 month old in a day...I am planning to stay in Piccadilly Circus...Just have a few questions...It would be great if someone could help me out on these!

1) Is car seat mandatory in London?( I will be taking a cab from Heathrow to the hotel in Piccadilly)
2) If so, can the cab guys get a seat, if we request it?
3) Cheap and good places to eat in and around Piccadilly:)
4) Is there any Tesco or similar store around P?

Thanks in Advance,
Lakshmi