1) Sorry. Never used Mathtype. Skip.
2) Not just writing math formulae actually. I've had use for it in
- Philosophy. LaTeX has logic symbols too (AND, OR, NEG) which makes it very handy here.
- Computer Science. Well, this answer is kind of moot as CS is almost Math. But my point is, you can use LaTeX to typeset pseudocode beautifully. Not surprising as TeX and LaTeX came from figures in CS (Knuth and Lamport, respectively).
- Pretty much anywhere. Believe it or not, I've used LaTeX to create papers in subjects in humanities. LaTeX has a very nice system for handling bibliographies and other features of an academic paper.
3) As I always say, if you know basic HTML (which even enthusiastic 8-year olds can master), LaTeX will be a breeze. Sometimes, though, the error messages LaTeX generates can get frustrating. When that happens, just google your error and you should be fine.