What is $H_2$ and $H_{\infty}$ control?
They are system norms, a metric that you can compare two different systems in terms of their generalized gain and spread. You can look these up no need for attaching physical motivation.
You don't. They are for assesing and using in the optimization programs.
Yes, but only if you know what you are doing. 
Can be. But again, they are for assessment and optimization. If you hit the minimum with a PID controller then it won't make any difference. But hitting the minimum is pretty much unlikely.
Unfortunately no.
see 5 (you can pull some tricks by bounding the nonlinearity with a linear system etc. or for very limited nonlinear operators you can apply them)
Not applicable. They are different things.
They are different norms. But both of them are norms. Read about them.
See the documentation of hinfsyn of matlab
Yes