Technically, the length of the aerial for any radio system (car, mobile phone, walkie-talkie etc), needs to be matched to the frequency you need to receive. Obviously there has to be a compromise as it would mean constantly changing the aerial's length as you tuned the radio to different channels, so the length the manufacturer chooses is already something of a compromise... So simple answer, yes, a shorter aerial will certainly work - you'll just get a lower signal strength and more drop-off of the signal as you move into weaker signal areas / further from the transmitter etc.
In and around my home area I can receive the stations I normally listen to without any aerial at all, (E.g. if I forget to put it back after going through the carwash!) but there are lots of 'holes' in reception where it cuts out for a while as I drive around (until I put it back).
Yes, the standard roof aerial simply unscrews. I've never looked to see if shorter ones are available anywhere - have you found one..?