Taking the OP's approach and expanding on that list
Fixed Order: Standard HQ - simple, fast, gets annoying when you want someone else to open the door or when a player gets "stuck" with nothing to do for several turns as their path is blocked by other heroes who are in turn blocked by monsters. Big advanatge to those higher up the order in terms of searching and looting chests. My first gaming group was my family, a parent was generally OK with playing the Wizard, never getting the gold (searching could mean double WM attack due to BDEW) but then they was nothing for the Wizard to buy, so they just took the hit of a duller game.
Choose Order: Simple but slower. If people just say "ok I want to go now" and "sure, here're the dice" then this is ideal. The problem is that some people will want to plan out every turn in advance, which means that the tactician becomes overly influential while the weaker players become less engaged. Even if all players actively engage in planning, this could add five minutes to every round. Another problem is contention, all is well when players agree, but when the first to move gets the chance to loot a treasure chest then all bets are off. My second gaming group was a group of friends so no one was willing to take a back seat, and we had added more artefacts for the Wizard to purchase so he wanted gold as much as the next hero! I'm looting the chest, no I'm looting the chest, no...
Fixed Order with a twist: Fixed turn order, but you can trade seats on Zargon's turn if there are no monsters on the board. This has all the disadvantages of a fixed order when there are monsters on the board (which is when it matters and those disadvantages apply) and dilutes the advantages of fixed turn order (it is simple but not as simple, it is fast but not as fast)
Fixed-Chose Hybrid: - players can choose but any contention is resolved through the fixed order (some have suggested fixed by MP value, but I don't agree as this represents magic capability / resistance and nothing to do with quick thinking), but you could either use an arbitrary initiative order EBWD or just use the seated order chosen at the beginning of the game/quest, although if you do that you would need a method of determining that, as all players would want to be first - players roll a die, highest chooses their seating position and so on
Random: - delay caused by randomising, needs randomizing device, risk of double EWP turn, breaks certain spell durations (lasts until the casters next turn, that could be the next turn)
Random Hero: - heroes uses random order but EWP plays last - delay in randomising, needs randomizing device, breaks certain spell durations - although it is better in that respect to pure random
Oops too many
Movement as initiative: - roll movement at start of turn, play in initiative/movement order, needs some method of recording the movement rolled, or 6 more dice, a player may still get "stuck" but this is now a result of a dice roll so is unlikely to be permanent. This allows monster and hero turns to be interleaved which makes the whole game much more dynamic but need a way of recording the movement roll (or lots more dice)
There is also "#+D6 Movement as Initiate" which is covered in this topic here but for the purposes of this conversation it is just a variant of Movement as Initiative although it needs less extra dice as only one is rolled per character
The Turn Order is not the same as The Fighting Order For example if you play fixed with the standard BDEW for example many groups seem to assume that means that your Barbarian leads the group through the dungeon followed by the Dwarf, then the Elf, then Wizard.
My second gaming group realised that doesn't have to be the case, they used an initiative based idea to determine the seating arrangement at the start of the game EBWD, but then allowed the Dwarf to lead through the dungeon (disarm bonus and tough enough to take the initial hits and plug a doorway as required), followed by the Wizard (to get into a position to "fire support" and heal the Dwarf), Elf (again to adopt a position for diagonal and/or shoot/spell support) and then the Barbarian (rear guard but could also punch through to clear a room as required) - I think that was the order and reasoning but it was a long time ago.
All that meant was on the first turn the Elf would typically search the starting room, and then move to one side of the exit door, the Barbarian would queue up one square behind the door, Wizard would take the other side, the Dwarf would then plough through and take the lead and people wouldn't overtake unless that was a reason to, like the Wizard or Elf stepping ahead or to one side to use missile attacks or the Barbarian moving forward to cover the Wizard or Elf or charge into combat.
This arrangement might not suit everyone of course, but my point was just that the Turn Order doesn't have to be The Fighting Order
Personally I think for the fighting order I would probably go with DWEB or DEWB