1: Vassal is a noun that describes a relationship that noblemen had. Technically every nobleman in Europe was a vassal to somebody else, except maybe the Pope. What I think you're referring to here are petty nobles (i.e. knights and squires and sometimes barons). So I'm going to proceed from that assumption.

2: Yes! Private warfare is called feuding and it was a HUGE problem, particularly in the later middle ages as government began to become more centralized, and the agents of the government tried to gain a monopoly on violence. Many nobles considered it their privilege to feud, and the representatives of the crown wanted to treat them like murderers. The conflict over the right to feud actually led the knights in Franconia (present-day central Germany) to revolt in 1522. (NB: There are several reasons for this revolt, among them religion, but the primary cause appears to have been the assertion of a baron's right to terrorize the citizens of a town he claimed to be feuding with)

3: Feuds could start for a variety of reasons. Most of the ones I've read about have their roots in a border dispute, but I've also read about feuds starting over a percieved slight to one's honor, and the mistreatment of a family member (usually someone feuding with their brother-in-law or son-in-law).

4: Feuding drops off in the sixteenth century, just about when duels (i.e. private single combat) start to become popular. I think there's something worthwhile to talk about here, but nobody has done the research.