Your buyer's agent should be talking with you regarding all your questions. Of course it doesn't hurt to get outside opinoins. Here is mine - It is difficult to predict what the banks will do. The banks are still new to this business of owning so much real estate and trying to sell in this buyer's market.
I had a buyer who put in an offer (conventional financing, strong pre-approval, 20% downpayment) over the asking price on a bank owned house in the Midway area. The bank accepted a lower offer that was cash. That bank was more concerned that the deal close with no delays or complications associated with financing - it was worth it to them to accept an offer $10k below the higher offer.
I have another deal in the works that the selling agent listed the house substantially below market rate to get a bidding war - he recieved 20 offers. My buyer came in over $50k over the asking price (conventional loan, 20% downpayment), the bank accepted their offer.
Of course there are many bank owned properties that the banks are taking less than the asking price - especially if the offer is cash with quick close. My advise to my buyers on bank owned properties is to make the offer as attractive as possible - cash deal if practical, if not - strong pre-approval, sizeable downpayment, as few contingencies as the buyer is comfortable with, quick close, healthy earnest money.
Best of luck on your deal - hope it works out in your favor.