Android is based on Linux; can native Linux applications be run on Android?
You can install chrooted linux distribution alongside android bacause android is based on linux kernel. If your phone is not rooted, you may use fakeroot (easiest way is to use Debinan nonroot app) even with GUI (with android X-server app or via VNC). If you have a rooted phone, you can install almost fully functional distribution.
I think the best performance and the least limitations you can achieve with Gentoo because all software compiles to your native arm architecture and it is the most flexible, but not the easiest. You may be interested in this post about installing Gentoo on android.
Yes they can if they're compiled under an arm linux first or using a cross compiler. Debian arm versatile works, there's also arm-eabi for compiling under x86 linux to arm linux.