Sorry, I disagree. No need for so many proposals.
PvE or PvP with no consequences is probably not necessary to discuss further. If there are no consequences (meaning no risk of dying like on an adventure), then definitely no reward should be given. Therefore no need to involve LPF Judges.
Putting on my DM hat, if I am not at least earning DMC, why would I do it instead of running a short adventure with 1-2 encounters? Answer, I would not.
As a player I feel the same way. I like doing encounters, but if there is no XP or GP involved, then I would prefer to keep my character eligible for upcoming adventures recruiting.
Bottom line with those two options, participants can do as Walking Dad suggested. Two players can go start their own thread and do whatever they want, leaving Judges and DMs out of it. Same thing if a DM is willing to run an NPC encounter without rewards, just go do it, no need for Judge approval like an adventure.
The other two issues can be combined for simplicity (pretty much using Systole's idea).
Proposal: Arena Combat Adventures (Players vs. Players and/or NPCs)
Rules: Requires Judge approval and possibly DM involvement. For PvP, must use EnWorld Dice roller, same as Arcane Row.
Where: In a sanctioned LFP IC thread. Either a shared one or separate, doesn't matter.
Style: Any, depends on participants and their agreement. Magic may be allowed or not. Fight can be to the death or not (accidental critical hit on someone with 1-2 HP not withstanding). Basically, this is governed by the DM and approved by the Judge, just like an Adventure.
Awards: As per CR, same as if it was an encounter & plus time. Cash prize to the winners, amount according to Table 12-2 in the Core Rules. DM includes this in the bout permission request. If the contest is to be non-lethal (i.e. with fists and such or paying -4 penalty with weapons) then perhaps the DM halves the CR for the encounter. Loser of a contest, receives time-based awards only.
Frequency: No restriction, since Judges/DMs are involved these are essentially mini-adventures.
Submission Process: OOC thread to request a Judge/DM. Once they agree then:
1. If PvP only encounter, then participants submit proposal to Judge via PM.
2. If NPCs are involved (even if PvP is being used), then DM submits proposal via PM to Judge. Proposal covers same things as an adventure.
I think that is it. While the badge system is interesting, it is a complication that is not necessary.
Other points with bearing on the discussion:
A. If someone challenges your character, you can decline. Simple as that. If you were ticking someone off at the time and then back down when they decided to take offense and challenge you, then you deserve to be a called a coward if that's how you role-played it. Essentially, the fact that the Arena exists doesn't change people's behavior.
B. Since a Judge would have to sanction any PvP encounters, time will pass if tempers are involved. Then, anyone shamelessly bullying will not get approval. As the activity is not actually happening in the DWI, then no need for Grog to bash heads either.
C. If the PvP contest is mismatched in numbers or levels (no reason it has to be 1v1, could be 1v2, 2v2 or whatever), then each side of participants should have an NPC or two assigned to make it more balanced. This will require a DM to be involved and a proposal submission to be approved by a Judge.
D. Why allow this at all? This is the biggest debate issue and it comes down to two simple points for me.
1. Provides an easier to get approved mini-adventure that will give a smaller number of characters something to do. If you can find someone with some free time (but not enough to do an adventure proposal) then you might not have to sit around in the DWI for weeks or months waiting.
2. If players/characters want to role-play conflict IC, there is a way to tell people to put up or shut up without breaking character. Without even the option, there is no threat of consequences for a character's IC behavior.
Fundamentally, I disagree with many of the people that think allowing PvP will cause people to have more conflicts and detract from their enjoyment in LPF. There are two sides to any IC disagreement, and both players are participating if they are posting. In reality, if you talk smack to someone, you deserve what you get. Being able to offend someone IC and not pay the consequences, is essentially a license to insult others and ruin their fun.