So far only GB has got it right!
1) Definitely stick with BMI DC for earning miles and you can earn PPS points at the same time. You should have been doing that before you moved anyway. No change when you move. If you compare the 2 programs it is quite obvious that it requires less miles to redeem using BMI DC. The only different as GB and Kiwi Flyer said is you cannot redeem on the 77W flights in J or F.
2) I usually redeem HKG-SIN in J and have no problems using BMI DC. Most *A airlines look at the same inventory award bucket. It is true that if you use your SQ KF points there are additional seats, but so far I haven't had to use that. I usually BMI DC to redeem on SQ.
3) For BA, stick with your UK address. Again you should compare your UK program to Asia Miles. In my opinion, the UK BAEC is still better than Asia miles. Don't change your address if not they will automatically transfer your account to Asia Miles.
4) Stick with your UK address or borrow someone else's address in the UK!