Melatonin only helps you to feel drowsy and ready for sleep. I've used it a bit for myself and a bit for when my children have trouble dropping off. I find it takes about half an hour to take effect and then doesn't last for very long so you have to get the window of opportunity right. Looking at my bottle of pills they are 300mcg and I'd take two. I wouldn't use it on a plane journey, but that's because I generally don't attempt to sleep on planes, for me it's night two when I need the help to drop off.
I wouldn't plan on driving as you don't know how it will affect you. Particularly if it is a long drive. I have done the short drive home after an 8 hour sleep. On the other hand you would be as dangerous if you had come off a red eye with no sleep for 24 hours