I put my gps in Dutch mode, so I actually knew this one already. I knew it would be useful!
Could I also say "Ik woon op de rechterkant?"
Unless you are Spider-Man, not usually. In this case, saying you live op de rechterkant would suggest you live on top of a literal right side of something.
Jap, dat klopt.
This is why you don't say "Ik ben op de trein".
Say "Ik ben in de trein".
Omdat als je op de trein bent, dan dat gevaarlijk is. En waarschijnlijk je bent in de trein, niet op het.
Would it be right to translate as 'I stay on the right side'
I would say that stay is more temporary than live. So f.e. you're staying in a hotel and living in a house.
Certain dialects of American English do use "stay" this way, but should probably be considered very colloquial.