What will happen to my bitcoins if Bitcoin splits into two separate coins? What will I own in that case, coin 1 or coin 2? Or both?
That depends on how you access your coins.
If you are using an account provided by a service then they get to decide for you.
If you are running your own wallet software, then you will technically have both, but the wallet software you are running may not provide access to both.
What if I don't touch my bitcoins for the next 2 years, leave them on a paper wallet (so to speak)?
Then you get to decide in 2 years when you choose what wallet software you want to import the private keys into. Technically you'll have both (if both chains still exist), but the wallet software you are running may not provide access to both.