I would say it is not normal to a bank to say "We'll consider a short-sale" in writing. Take it as an omen from God! Kidding aside, go with the flow.
Do your numbers and figure out what you want to pay for it, then anchor your bid some amount below that, maybe 80% - 90%. Too low an opening bid and you don't look professional. Too high and you have no room to negotiate. If you can include inspections and/or contractor quotes to confirm why your offer is what it is, more power to you. There is no rule about getting quotes and inspections before you are in contract, with owner permission, of course.
If you really want your offer to have power, put the funds into escrow and give the bank the number to contact. Tell the bank in the cover letter for the offer "I have X in an escrow account with X company. I can close in 7 days." Notice you are not making an offer "subject to inspections" because you already did that and you want to so save the bank time(money). After the bank confirms your funds are in escrow, they will know not only do you have the cash but that you are a power player who doesn't waste time in making things happen.
To up your game a notch, deliver the offer in person to the manager of the bank and shake his/her hand, look them right in the eye and use Jedi mind tricks to convince them you are the real deal.
Good luck and report back!