1) yes
2) yes(I don't know which, for most consoles, the only way you can legally get the APIs and/or the actual libraries is to become an official developer for them)
3) no, some libraries(like SDL and Allegro) work on many platforms and OSes

depends on how you make the game. if you make one part of it just handle the platform specific stuff, then you simply change that for the new platform, not the whole game

Developing for a console can be very different than a PC in some respects, while in others it's the same thing. The biggest differences being you usually talk directly to the hardware on a console, and you can rely on there always being the exact same hardware with the exact same specs for every person who runs the software. This is not counting all the design changes you have to worry about, and the often limited resources of a console.

Most consoles don't provide anything resembling an API as you are thinking of (especially for the graphics hardware), work is done by talking directly to the hardware. Usually the closest thing to an API are 3rd party middleware libraries. The exception to the rule is the Xbox, which uses a slightly modified version of directx.

You also have to think about the screen resolution. While your game might be run in 800x600 or 1024x756 on a PC, it'll be running in what would be considered around 600x480 (actually it's smaller and there are other differences). So textures have to be remade (resized) so it looks good, but also to preserve memory. What I'm saying is, even if you manage to make your engine compile and run on say a PS2. Your artist is not going to thank you for it ;)

Now I just ignored all the fun bugs that will show up on a PS2 and not on a XBox, or a PC. So be prepared to stock up on Jolt Cola.

Also, have you tried a ported games? Everybody just loves ported games! They somehow end up sucking on a platform that they weren't made for. Some have managed to make OK ports, but most fail. That's because games are played so differently on a TV with a handcontroller while sittin on your couch, compared to sitting in front of your computer screen using your keyboard, mouse and maybe headset.

I would go as far to say that making your engine (code, not game) run on several plattforms would be the least of your problems.