Seems like the best of the suggestions here would be a combination.. with a bit of a twist:
1. Make the drop-down menu bigger. People are often doing this in a hurry. Some zergy-types don't do it at all because they begrudge the couple of seconds it takes. But it's often done in a rush, regardless.
2. Add a confirmation. Yes, it's yet another half-second, but would save a world of pain.
3. Add a, "Put this item up for roll," option to the menu to automate rolling.
- Everyone remaining in the quest gets a prompt (NOT over the chest.. in General and/or Party Chat is probably the best place) to choose to 'Roll' or 'Pass'.
- Ideally, it should have the item link as though it were Ctrl-Rclicked with the "'Roll' or 'Pass'" option beneath in case the item link goes beyond the chat window's right side.
- It would need a timer to automatically pass the item to the winner after 10 or 15 seconds of awaiting clicks (Maybe 10 sec in quests and 15 sec in raids?). Show the countdown beside the "'Roll' or 'Pass'" option for each item.
- Might be best to add 5 sec to each item roll timer for each additional item simultaneously up for roll, to give players more time to peruse the options.
4. If the owner leaves without passing a named item, it is immediately, automatically put up for roll among remaining players in the instance if the quest is run on Reaper. On lower difficulties, it waits 1 minute before doing so in case the owner chooses to re-enter for the item. People running below Reaper are seldom zerging; people running Reaper can't re-enter.
I think that would cover all the issues pretty thoroughly.
One additional potentially helpful change (perhaps enabled in Options): Show only your own loot in the main chest window; show everyone's named loot in a separate window so it doesn't get scrolled offscreen by trash loot. Possibly add separate 'Loot' buttons for each, along with a 'Loot All' button for those who'd rather not click twice.
And, ofc, while they're at it they could put the 'Reroll' button at the top of the window so it never gets clicked accidentally again.