A concert uke generally has 12 frets up the neck and four more up to the sound hole on the body. If your fretboard doesn't extend past the end of the neck I guess it would only have 12. Generally it is the size of the body that gives the instrument it's sound, not the scale length, hence long-necks.
As for the buzzing strings yes, when you strum too hard you can make the strings buzz. *more physics* For every action there is an equal and opposite reaction, think of putting a rubberband between your fingers and snapping it against your arm, the further you pull it in one direction the harder it comes back the other way and the further it goes.
You could get a little more out of some stiffer strings, for example a softer string like an Aquila Nylgut will probably buzz more when you play real hard than something like a GHS black or a D'addario guitar string.
Another thing that might help would be to strum with a soft pick-the soft wedgies are great.