When things get fishy and you are not sure that the offer you are so into is legitimate, there is one hard, fast, unbreakable rule: DO NOT PAY ANY MONEY!
Secondly, feel free to pester. Let the potential landlord know you are uncomfortable with this situation. Do not rest until you get a working phone number and address to reach the person, not to mention the verifiable address of the rental property.
If it is at all possible, acquire the services of a broker to act as a go between. He or she will be able to ascertain whether this rental offer is legitimate, and it will save you from getting duped into anything.
Keep other properties in mind. Sure, I am in love with this house, it suits all my needs, and it is affordable, but I know I logically have to realize that there are other places available out there. I cannot hang all my hopes on this one property; I simply have to hope it turns out well.