Ansys and Competitors

Try looking up Finite Element Analysis software.
MSC.Nastran, Abaqus, IDEAS, Algor, ...
Anywhere from 1 day to years, depending on how through one wants to learn them who is trying to learn them.
Don't know about easily, but it has to help a lot.

Jeff answered some of these, but I'll try to provide some additional insight...
Jeff listed several good ones.
Depends on who you ask and what your judging criteria is. All of them have some strong points, and all of them have some weak points. Price, processing time, accuracy, required training, etc...
Not much of a clue... try checking their annual report.
Well, to really learn it, you need to have ATLEAST a basic foundation in mechanics and the appropriate mathmatics. If you can't grasp the concept of a node or a mesh, then you're screwed. After that, you probably need a little CAD experience to understand computer aided design. After THAT, you're probably looking at a week (for novice understanding) to years (for expert understanding).
In some ways, yes... most concepts (nodes, meshes, constraints, etc) are common between FEA software. It's mostly just learning a new interface if you switch software.
Depends on what you mean by "big". But pretty much any mechanism that's of any importance has probably been subjected to an FEA. Space vehicles, cars, cell phones, buildings, etc...ALL commercial and military aircraft made within the last 20 years, at some point in time, have been subjected to an FEA analysis, whether it be at a component level or a top assembly level. It's safe to say that FEAs are a huge design tool that the general public doesn't really know or care about, but is highly influential. On a side note, FEA's help alot... but, we made it to the moon without them.
Dave