- Should I cut the tops off so they look like they do in the store or leave them on?
You can braid them, which requires leaving the tops on. If you are going to cut the tops off, don't cut more than an inch.
- Should I remove any of the papery outside layers, for example if they are damaged or covered it dirt I can't easily remove?
Don't remove any layers. Don't wash them. Gently brush off dirt when they are fully dried if you must. You want the onion intact for storage. Any that are damaged should be used right away -- these won't store well.
Any that are thick-necked, spotted or soft will rot if stored normally; they need to be either eaten within a few days of lifting or frozen. [From a comment on this post by Mancuniensis.]
- I live in a hot climate so there are no cool locations below room temperature except the refrigerator or freezer. Is a fridge well ventilated enough and does it have low enough humidity for long-term storage?
Fridge humidity is too variable. Your produce drawers may be around 80%, but if the fridge is left unopened for a period of time it can become very dry. Bottom line is that they are likely to spoil more quickly -- and they will make everything in your fridge smell like onions.
- Do they need to be somehow cured or prepared after harvesting but before storing?
Onions should be cured for a couple of weeks. Dry them on a rack or other place with good circulation out of direct sunlight. Turn them a couple of times so that they dry evenly.
- Is there anything better than a mesh bag? For example, wrapping them in foil.
I had never heard of wrapping them in foil. If you store them in a mesh bag (really, with any storage in general), be sure to check frequently for spoilage -- if one of your onions spoils in storage, it will cause rapid spoilage of all of the other onions nearby.
Given ideal storage, how long can I expect my onions to last, and is there a way to tell if they're about to go bad so I can use or dehydrate them?
Mine never last this long (I don't have ideal storage), but according to books like "Root Cellaring" by Mike and Nancy Bubel, onions stored under ideal conditions can last from a late summer harvest until the following spring (6-8 months).
Since you don't have a good place to store them long term, I'd recommend using them or dehydrating them now. (You don't want to dehydrate onions that are about to spoil -- it will not make a good product.)
This page describes freezing as an alternative storage method:
you can salvage them by peeling and pureeing them in a blender. Pour the puree into ice trays, cover them with plastic (so the odor won't affect other foods) and freeze them. After the onion cubes have frozen, transfer them to a plastic bag in your freezer.
and for whole onions:
Peel and wash the onions and blanch them in scalding water until the centers are heated (three minutes for small onions, seven minutes for medium to large ones). Cool, drain and put the onions on cookie sheets, and place the sheets in the freezer. After they're frozen, put the onions in a plastic bag for convenient storage
with this caveat:
For the best flavor, use frozen onions within a month or two.
You could also pickle them.