When there is money left from a forclosure sale or auction after the
debt, accrued interest, and other expense have been met, does the
borrower get the remainder?

According to these sources, the answer is yes in most circumstances. 
There appears to be an exception, however.

"Likewise, if the foreclosure sale results in a surplus, the mortgagor
is usually entitled to seek payment from the mortgagee for the amount
of the surplus." CHRISTOPHER BEBEL, ESQ. PC (2005)
http://www.chrisbebel.com/CM/BankingandFinanceNews/e-newsletter.asp

"Previous owner gets the overbid."  By Mark, HighWired Internet
Innovations Inc(June 14, 2005)
http://www.all-foreclosure.com/forum/messages/17550.html

"13. If lender overbids. A lender who mistakenly overbids at
foreclosure sale may be able to secure equitable relief and avoid
having to make payment to an inferior lienholder or the mortgagor for
the excess under some circumstances. McClure v. Casann Claire
Apartments, Ltd., 560 S.W.2d 457 (Tex. Civ. App.--Beaumont 1977, no
writ)." Balcom Law Firm, P.C. (2005)
http://www.balcomlaw.com/publications/foreclosure_manual/46.htm#iih13

Search terms: foreclosure surplus Texas

Sincerely,

Wonko