Game Developer Eligibility

First of all, I am glad that you are very clear about what exactly you want to do and it doesn't matter if you got 60% marks in class 12th. I don't know much about Institutes offering game development courses, so I am skipping any suggestions regarding which Institutes to join. But, here is what I got to know while trying my hands in game development as a hobby. Game development is not just programming or just character designing or level designing, you have to be relatively good at everything, if you truly want to make production level game. And, thankfully there are lots of tools and resources for game development and I would say it's quite easy nowadays to jump into game development. I tried game development in JavaScript thanks to great open source gaming libraries and in Unity 3D. There are lot of tutorials and documentation floating around on internet. And, if you are relatively comfortable with basic programming, then you can easily become well versed in either one of the game development environment. Just choose which technology stack you want to use for game development. I would suggest you to use Unity 3d as you can port your games to many platform like from android to IOS to Web Browsers.
So, download Unity 3d, or favorite JavaScript gaming library and if you want to go native way, then download Android SDK or Xcode for iPhone, so basically a development environment.
Then join one of many Udemy tutorials in the development environment you chose and start coding and creating. Don't give up if things seems complicated at some time because they are. At the end of the course, you will have a very good idea about how to develop games and once you get your basic game development concepts clear, then it will be relatively easy for you to transition from one platform to another.
And, talking about making money through games, there are many ways to monetize your game, you can try Ads in your game or follow In App Purchase strategy or outright sell it on app store for the price you think it's worth or work for game development studio or as a freelancer.