Stream cipher - Many time pad
No usually the shorter message doesn't get padded, you can just not decrypt or at least analyse the last part of the longer message. If the key is not even the same size as the message then you are dealing with a variation of the Vigenere cipher, not with a one-time-pad.

What happens in practice is of course dependent on the situation. But if the key is reused of an OTP then the security is lost, and usually in a rather catastrophic way.