Having the home listed with a Realtor is not going to prevent the bank from setting a foreclosure date. The best advice I can give you is to contact a lawyer and have them get involved and call your state Attorney general's office and speak with them. Too many banks are foreclosing illegally on homes that they cannot even prove they have title to. The States' attorney generals are well aware of this and are buys filing suits against these banks. I have no idea of the specifics of your situation, but this is the best advice I can offer you or anyone else in a similar situation.
Yes, they will. However....it has been my experience with Short Sales (which I'm going to assume this is) that if you get an offer at least a week before the Trustee Sale/Auction, your Realtor can submit it to the lender(s) and ask the negotiator to postpone the Trustee Sale/Auction. Make sure your Realtor does this as it's not automatic. And, your Realtor will need to keep in constant contact with the negotiator because most Trustee Sales/Auctions are only postponed for 30 days. Your Realtor needs to keep on top of the situation because I have also heard of situations where the property had an offer in but the Trustee Sale/Auction still occured - the 'right hand' didn't know what the 'left hand' was doing.
They sure will. They won't continue to put off the inevitable because the house 'may' sell. I've seen homes that were under contract go to sheriff's sale a couple of days before the closing was scheduled. Sounds crazy, but it happens quite often.
In my experience they still will. But that date does not mean that the action will be implemented. I believe they do it to as a fail safe in case they determine that the appropriate steps are not being taken to actively market the home and secure a Buyer. I have found that the date often gets moved, postponed or eliminated as the transaction is progressing.