Legal Question in Real Estate Law in California

Yes. Anyone with the necessary money can bid and win -- the borrower, the lender, junior lienholders, even the trustee, although a trustee bidding for himself/herself/itself is discouraged because it can give rise to suspicion of unfairness or other wrongdoing.

As long as their money is good.

Yes, but you should be aware that placing the title back in the original borrower's name reinstates the deed of trust. "Title acquired by the mortgagor subsequent to the execution of the mortgage, inures to the mortgagee as security for the debt in like manner as if acquired before the execution."

In Barberi v. Rothchild, (1936) 7 Cal.2d 537, a fee owner executed two trust deds. The owner the defaulted and the property was sold under the senior trust deed to the owner's brother. The former owner filed a petition in bankruptcy and recived a discharge. He then reacquired the premises from his brother. The court agreed with the beneficiary that the second deed of trust reattached and could make the property subject to sale under the power of sale in the second trust deed.