How do I change my Gamertag in games?

I need some help or clarification on if I can change my Gamertag in the few games that I play on my Android Device. For example, my First Name last Initial shows up along with my Picture in all the games when I log in. However, I would like it to show the picture but have a gamertag or use a different name other then my own. Real Racing 3 is a prime example. There are a ton of users using Gamertags/Avatars, however, you can only login using Facebook, some chinese website and Google +. There is no where in there to change your name. I have looked everywhere. Also if you look at your trophies I would like it to show my NEW USERNAME. Does anyone have an answer or suggestion? Can I create a second Google+ account and not lose all the points I have acquired? I have searched everywhere for months actually as the answers to most questions are usually found with a simple search.