1. What else can be used aside from Unity? Advantages/disadvantages? Options are always nice.
You can always just write your own engine. AIW was in a custom .NET/SlimDX engine Chris wrote, until version after version 3. Using something like Unity saves you time but you also learn a lot of important stuff writing your own. That said, if you want to get to a playable game quickly, picking an existing engine will generally be a better bet. And commercially it's nice to be able to count on someone else for the cross-platform support and sensible low-level graphics code, etc. Though I won't pretend there aren't times where I wish I had the nearly-total-control of our old engine.
As far as other options, there's also XNA which I wouldn't suggest for commercial projects (Unity gives you a lot more platform options and is a lot more flexible about how you do some things and is far easier in the actual distribution phase) but I think it could be a good one to learn with because you don't have to deal with as much external tools (at least as far as I can see; visual C# express is free and is all you need) as opposed to learning the Unity editor (which isn't bad, but it's another piece).
2. In Unity, can you actually make 2d games?
Absolutely, just draw each sprite as a textured quad (two triangle stuck together to form a rectangle, and then a texture drawn over them) in front of a single camera with an orthographic projection. Chris did that code for us, I could ask him for details that someone else could use to do the same thing. To a large extent, though, it would be good for one of you to figure it out so you understand what's going on. That won't be the fastest path to a game, but you'll learn.
But yea, support for literal 2D drawing in the old way of just writing pixels to the frame buffer or whatever... yea, not really
It can be closely approximated (iirc) using the GL class in Unity, iirc, but it's not a good idea. The video cards are optimized for 3D tech, so cut with the grain. The results can look just as 2D as it did the old way.
As far as a relatively simple idea, I'd say something top-down that runs in real-time (turn-based is actually somewhat more complex because you'll want to be able to do fluid animations and probably fluid changes in the actual simulation model, so starting in TB can actually lead to some counterproductive architectural changes). That ship-shooting thing you mentioned would work, or whatever. As far as something that's simple
and fun, well, one step at a time