Trying to upgrade. I can open my wallet in Multibit classic without entering a password. I have never needed a password to open it and don't remember ever setting one up. I certainly have none recorded. But I can't export a key without entering a password. I tried sending the contents of the wallet to a new wallet in Multibit-HD but I don't know how much to send to allow for the transaction fee. There does not seem to be an "empty wallet" option.
MultiBit developer here. You should be able to simply spend from your Classic wallet into your new HD wallet. Don't try moving private keys around - that's really easy to get wrong and is one of the reasons we wrote HD in the first place.
There are full instructions on upgrading on the MultiBit website: https://multibit.org/en/help/hd0.1/how-to-upgrade-from-classic.html
Well I'm not totally sure about the password situation but you can easily send a 1 cent transaction fee and send the rest of your bitcoin to another wallet if that is an option. You don't even technically need a transaction fee it will just take much longer for the transaction to be confirmed if you do not send a fee.
1 cent currently is ~ 0.00004 BTC