Neither are palms. The first is a Xanthorhoea species. There are several species from different climatic areas. Bit hard to tell what the problem is but yours does look in bad shape. Over watering or poor drainage could make it look like that.

Is the second some kind of tree aloe?

Thanks tropic!! So there probably isn't a hole in the bottom of #1's pot.. how long should I wait to water it? Once a month? I cringe at the thought that I damaged it by over-watering! :( Dang it!

If there's no hole in the bottom of the pot then get it out of there as soon as possible. Unless it's a clay/cement pot and you have a drill with a large masonry bit that you can drill a lot of holes with. Otherwise, you have to repot into a suitable pot using a course sandy soil mix. The shape of that pot means you'd have to carefully break it to get the plant out.

Here's a photo of some flowering in habitat. You can't see the trunks on them but they'd be quite a bit taller than yours. They are a beautiful plant and well worth trying to save.