I have developed some Unity3D projects using GitHub before. So to answer question 3 and the last part of 1 first, yes Unity projects use a file-system architecture that is perfectly compatible with GitHub and once your used to it it is a great tool for team development.
Answer for question 1:
GitHub is just a name brand for a centralized version control system and there are other brands out there with similar offerings such as bit bucket. Google this term for more info. also look into distributed version control as well.
In all honesty though, if your new to developing, the product you will be making will most likely not be of much interest to other people on GitHub and your public repository will probably go unnoticed. If you believe that what you are creating is of such great value it needs to be kept secret, then investing a few dollars a month in a premium service is recommended anyway.
For other options, one would be to set up a central Git repository on a server (or one of your home computers) that you or one of you project mates is running. This might be a more complicated method but you would learn a lot of other useful things along the way.
Answer for question 2:
See -https://guides.github.com/activities/hello-world/- for github's intro tutorial. Also Youtube has some decent offering if you search for how to use Git Hub.
It can be a little daunting to work with something new and attempt to understand the documentation. If you are planning on getting serious about development though, especially in a corporate setting, you need to learn GIT and practice reading and understanding documentation.
Good Luck!