It seems clear that there is a prevalent custom among Jewish men to have beards.
Question 1) Is there a source for this?
Question 2) Is there any difference between why some keep it short while other never cut it?
The reason is partially because there is negative transgression to cut certain parts of ones beard with a razor, and by growing a beard, this won’t happen. Another reason is because there is a Jewish look to it.
The reason some people don’t cut their beard at all is for kabalah reasons and not because of halachic ones. See Beer Heitiv Y:D 181-5.
Yaaros Devash 1- pg. 68b, 72b, MIshnas Yosef drasho 113 for other reasons, Livushei Mordichai Y:D 99