We are in the process of becoming a 1031 Qualified Intermediary. Do I understand correctly that, so long as we are opening the account in our name, we are the customer and we have no responsibility to CIP or collect Beneficial Ownership on any third parties? What about OFAC?

I would be engaging my legal counsel and a qualified CPA in the discussion of what legal responsibilities the bank may have when acting as a qualified intermediary. I assume this is being done through your Trust Charter.

Funny thing, they have engaged me on the BSA side

So, the bank itself is going to be the QI? Would not whoever you were acting as QI for be a customer when you sign the QI contract with them?

(a) Account. For purposes of §1020.220:

(1) Account means a formal banking relationship established to provide or engage in services, dealings, or other financial transactions including a deposit account, a transaction or asset account, a credit account, or other extension of credit. Account also includes a relationship established to provide a safety deposit box or other safekeeping services, or cash management, custodian, and trust services.

Yes, we are.
My research has given conflicting answers and now my mind can see both sides:
I read the regulation you posted and think yes.
This is dated but says they would not: https://www.bankersonline.com/qa/cip-1031-exchanges

Also, in all the research I have done I have found nothing regarding OFAC....my guess is to OFAC both sides of the transaction.

my guess is to OFAC both sides of the transaction

I would be. When it comes right down to it, since this is a trick to avoid Federal taxes, I would think that the bank would want to know exactly who they are dealing with all the way around.

as always, there is no requirement by the US Treasury for any bank to perform any type of OFAC checking, they simply make it illegal to provide payment to anyone on the various OFAC sanction lists. your bank makes a risk decision whether to scan this list or not. failure to scan when the US Treasury comes knocking will not result in favorable outcome for your bank.