How do I root LG Optimus F60? - MetroPCS LG Optimus F60 4G with 8GB Memory No-Contract Cell Phone

Maybe a few, but keep in mind stability or lack hereof and read a few forums on a few sites like :
http://forum.xda-developers.com/

http://droidors.com/2015/01/01/how-to-root-lg-optimus-dynamic-2-l39c-using-one-click-framaroot-apk/
and see all links an take responsibility for whatever method, ROM you use. I am simply providing you resources I found.

http://forums.androidcentral.com/lg-optimus-series/406108-rooting-lg-optimus-dynamic-2-lgl39c-tracfone-stock-rom.html

http://update-phones.com/android-update-for-lg-optimus-dynamicl38c/

http://overstandingvision.weebly.com/majkbot-hak-blog/lg38c-turbo-stock-rom-rooted-backup-format-for-lg-l38c

https://www.youtube.com/watch?v=yH3qKLXqdSg

Check to see if your model will work with an MHL adapter. MHL What is MHL How Do Use It

Check this site:
http://sim-unlock.net/unlock/LG/
http://sim-unlock.net/unlock/LG/

I assume you're talking about the Optimus G? If not, please provide specific model number. Anyway, it's been awhile since I've had that phone, but when I did, you had to have a computer to root it. I don't believe that has changed. Step-by-step instructions can be found at the following links:
GUIDE ROOT BL Unlock Install Custom Recovery LG Optimus E970 1 3

LG Optimus All Things Root Guide Android Forums

Rooting for LG Optimus sprint Updated 3 13 13 Android Forums

Suggest send phone along with proof of your ID to LG Optimus and have them recover your codes.

No, Rooting wont solve the problem. Its quite easy to unlock LG devices for other network providers. All you need is the correct unlock code for your LG Optimus and you are good to go.
Follow the link to unlock LG smartphones

Youtube.com type the name of your phone and root.