In theory, you'd only need to add SDL_image and SDL to your linker.
If they are installed, you can use -lSDL2 and -lSDL2_image
If you are compiling it yourself, this command will probably work (given you have autotools and build-essential installed):
./autogen.sh && ./configure && make && sudo make install
It will compile and install SDL2, probably under usr/local. It would also work for SDL_image and SDL_mixer...
The libpng runtime come in every linux distribution I've ever used (save arch), so don't worry about it. Also, if someone is using an Arch or Gentoo or anything that doesn't come with the runtime libpng, they probably know what they are doing (and will understand the missing shared object error) and will be able to fix it themselves (basically a call apt-get or pacman).
But yes, unfortunately, linux does not link statically. This is a major pain when considering this kind of dependencies, but well, them's the breaks.
Also.. since SDL 1.2 is more portable. I predict it will actually outlive 2.0.
SDL1.2 is not necessarily more portable.
You can't use it on iOS in a practical manner, given the legal restrictions the LGPL license imposes. SDL2 is zlib, that is way more usable from a developer point of view.
There are also the more obvious reasons; SDL2 is fully hardware accelerated, it has a better API, supports OpenGL ES and 3+, to name a few.
So, I'd bet all my programming books that SDL2 will outlive 1.2 easily, the same way I did saying XNA would be discontinued.
@edit
SDL1.2 is already marked as a historical version, and looks like it hasn't been updated in over two years.