Can the bankruptcy trustee try to shortsale your house before the stay has been lifted instead of letting the bank go through the foreclosure process?

Yes, but this is very rare. A Chapter 7 bankruptcy trustee will generally not seek to sell real property if the mortgage and other liens exceed the value. The Trustee's job is to liquidate assets for the benefit of unsecured creditors. If a Chapter 7 trustee were to sell a house that is upside-down, only the secured lien holder would realize anything from the proceeds, the trustee and unsecured creditors would get nothing. It would be an exercise in futility

However, some aggressive Chapter 7 trustees have been known to try to force a short sale and/or stall a foreclosure in order to negotiate some sort of payment for the estate.