They do. In fact, they typically hold a little more, 80 cf vs. 77 cf on an AL80.
Sure. I'm assuming that you're asking about singles on recreational (no-deco) dives.
New cylinders --> Choose an HP100, HP117, HP120, or HP130. I would guess based on your post that HP100 will still be too small, see "capacity notes" below. Of these, HP100 and HP120 are 7.25" diameter, same as AL80s and HP80s, while the others are 8" diameter. HP120s are long and work out well for fairly tall people, if you are not especially tall, you will prefer an HP117 which will give you the same capacity in a cylinder with less length.
Used cylinders --> Also consider the LP95s or LP108s you mention if you can get them at a good price for their condition.
In addition, there are HP149s out there (new and used) and LP120s (used only), which are widely considered too large and too heavy for routine use, but which a handful of relatively large, athletic divers prefer. They are 8" diameter and the same length as HP120s.
Capacity notes
It would help to know your SAC or at least your approximate height and weight.
In very general terms, for air dives within recreational limits, an HP100 will provide sufficient gas and substantial reserves for most people. For shallow dives, most people can easily get an hour out of an HP100. For deeper dives, following the usual pattern of a two-tank dive with an hour surface interval, NDLs will be limiting on the second dive, for most people and most profiles.
If you are diving nitrox then gas capacity, not NDLs, will limit the duration of deeper dives, especially if you are only making 1-2 dives a day, and larger cylinders will allow you to dive longer.
Among HP117, HP120, and HP130 cylinders, the difference in capacity isn't especially significant, and you would want to choose cylinders that fit your body shape and are practical for you to handle topside.
Stages and doubles
Even on a no-deco dive, some people bring a stage bottle. There are various strategies for using it. You can use it as a reserve and reduce the amount of reserve (and therefore increase the gas available for the planned portion of your dive) in your back gas. You can also plan to use the stage bottle towards the beginning of the dive and switch to your back gas as appropriate. Usually an AL40 or AL80 is used for this purpose.
Some people (including me) use doubles for certain no-deco dives, for reasons of capacity, redundancy, or both.