My policy is, until I see a text editor as good as
TextEdit on another operating system, I won't even
consider a comparison between the two.
1) Nothing is as good as TextMate.
TextMate does some awesome stuff, from what I've heard, but it's much more focused on coding. TextEdit is more general-purpose, which is why it's my standard. Until another OS spends as much time making a kickass all-purpose editor, it probably hasn't got the level of detail that I want in my operating system.
I've never used LyX; forgive me if this is a dumb question, but is it available for Ubuntu? I might boot up a virtual machine and give it a try if so.
It's also available for Windows/OS X so you might be better off just trying it on your primary OS.
I don't think you're going to find the kind of simplicity you enjoy in TextEdit though.
Personally I use text-editors e.g. vim, for every kind of writing I do, code or not.
Plaintext formats like http://eleanor.rubyforge.org/ (for screenplays) and Taskpaper (for planning/todo lists) are really helpful for this sort of thing.