The bank does not stop the foreclosure process just because you have listed the property. You or your agent should be in constant contact with the bank. When an offer comes in, it should be presented to the bank who will either accept or turn down the offer. However, this could take up to six weeks. Make sure that your agent in knowledgeable and seasoned with foreclosure sales. If not, it is time to find another one.
Keller Williams Main Line Realty
If your property is just on the market, I am pretty sure they won't accept the offer and take the listing off the market.
It depends on the situation. I would make sure that I am represented by a good negotiator (not the listing agent...your own representation is best). You never know what they will do until you ask.
Depending upon the market, they might give you a "right of first refusal". At least that keeps you in the race. If not, arranging a HELOC (Home Equity Line Of Credit) is usually a relatively quick process, assuming you have a good credit score, documented income, etc.
I am a believer in that principle if you do not ask, you'll never know.
Best of luck!