Does the Avois address need to be the same as the BAEC address to transfer points from one program to the other?
Nope. I opened an Avios.com account to try to transfer BA to IB. I was hoping it would circumvent the rule that the IB account must be 3 months old to accept transfers from BA. I used a UK address and had no problem transferring from BA to Avios. Unfortunately, I was misinformed about the 3 month rule. I also applies to transfers from Avios to IB. So, I transferred from BA to Avios and then back to BA without addresses matching.
A question for the OP -- is there a way to book eurostar on line from avios? I would prefer not to have to make an international call to book if possible.