Rooting does clear up the touch screen problem in some cases.
Ok so if you want to root your G2, here are the steps.
1. Go to settings and make your phone accept unknown apps.
2. Download Towelroot and install it on your phone.
3. Download Autorec and install it.
4. Now install a kernel you want.
5. Download a rom that matches with your kernel. It can be a AOSP rom, or a G3 rom. Run both and see what you like.
6. Always look and read each of the rom posts carefully and download GApps.
When downloading a rom, you should do a factory reset ALWAYS so information doesn't get mixed up.
I know this is a short list and im missing other items but there are tons of posts on how to do these steps in much more detail. And if you need help, ask away.
LG G2 - LiquidSmooth Rom
If I helped you in anyway, y'all know what to do.