In response to the cramping: Aside from stretching and warming up, you'll just have to just get used to riding. Eventually you'll build up the muscles that you use for cycling and you won't ache as much when riding. 13km is a decent distance for just starting.

In response to the bike: The seat is supposed to be high enough so that your leg is almost fully extended when your foot is at the bottom of the pedal stroke. If your seat is at the right height, you shouldn't be able to sit on the seat and put your feet on the ground at the same time.

check out http://www.bikexprt.com/streetsmarts/usa/chapter1a.htm for an explination of how you start/stop (and a bunch of other things)

Your bike might have been slow if you were either in a really low gear or a really high gear. The best way to build up speed is to start in a low gear (where it's easier to pedal) and slowly shift up to the next gear as you go faster (and then shift back to a lower gear when you come to a stop).