A PvP system needs to be balanced by a
System
of
Law before it can be used fully for RP purposes. People who then commit illegal PvP actions and get caught are
SoL.

As per the questions, PvP can be RP in the current system (no changes to the code) if...
... it is a fight between gladiators for a money (or other) prize.
... it is a criminal attacking a victim OUT of the public eye. Since you can not loot another player forceably, this could only be to demand money (if agreed on before hand) or because the criminal is a psychotic killer (if the victim is killed) or a thug if just beating someone up for whatever reason. However, this option is only viable if the criminal is willing to Roleplay out the consequences if caught, which are generally not much fun as per the current game mechanics.
... it is used to RP out training between students and teachers. RP only, as it really does nothing in the current system of only being able to train off NPCs.
... both sides are willing to RP the consequences.
Question two. There are many many other scenarios, but all of them require the players to carry through with the results. Stabbing someone in front of a guard leaves no 'mark' on your character. Those who are killed and rush back out of the DR have no ill effects after the curse wears off. PvP can be pretty much ignored by RP, which is why it is not used much. Sometimes PvP is used to resolve conflict, but that sometimes only lasts until the player gets back out of the DR. PvP as the system is now, in essence, has no power over roleplaying out of full PvP zones, and roleplaying has little power inside full PvP zones (if those inside the zone are non-RPers). In effect, the only way for PvP to have an effect on RP is if those controlling the PvP are, in fact, roleplayers.
Question three. First, the positives. Roleplayers often use PvP as a basis for creating storylines and resolving them, as do GMs. Without PvP, many of these stories and events would never happen, and without RP, the PvP would not happen in such a positive manner. In other games, the PvP is used to 'pwn' the other team and the rub it in. PS does have a better effect on PvP in that it is used to bring the community together to solve issues.
The negative. RPers who do not use mechanics in their 'PvP' often frown on, or make outright demeaning remarks on those who do. These 'textplayers' are often louder than others in proclaiming what is 'right and wrong' and have made a lot of people who enjoy both RP and PvP feel as if they have to hide the second part to avoid ridicule. In doing this, they have made PvP taboo, and as such, made bringing PvP into more prominent roleplays harder to do. This also makes it harder for people who do not know much about RP, but like to fight to get into the RP side of the game.
For the last, how should it be used? As anything, it should be used as a tool to promote, initiate, continue, or conclude a RP. It should play a part anywhere where conflict is called for (rather than textfighting). However, given the mechanics of today and a lack of a system of Law, it is hard to use it as such. Given no recourse for most PvP actions, a lot of people A: refuse to accept challenges into their roleplay. B: Refuse to roleplay the consequences of the PvP. Until both of those issues can be addressed, PvP will not be used to its full potential, and there will remain a gap between the 'two sides'.
By the way, I am a PvPer as well as a Roleplayer, just not in PS, as I don't like the mechanics of it (in other words, PvP is boring to me in PS, and gives no benefit to training). I also never 'RP' fight for the same reasons.