Legal Question in Real Estate Law in California

Yes.

I don't see any reason why not. The agent probably should disclose the agency relationship and the principal's name, to avoid possible personal responsibility, and would need to be prepared to meet the trustee's "payment in cash" requirements. The agent's bidding instructions should be made very clear to avoid over- or under-bidding mixups. Finally, the principal here should probably check with the agent's broker to see if the proposed bidding project meets with the broker's approval.