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So, the easiest things to see is that anything with a square is non-linear. I'm not certain what they mean by "semi" and "quasi" linear, to be honest.
The orders are the highest derivative in each equation, although I need to find my book on inseparable equations to be sure whether it's with respect to a single variable only or not (I believe it is).
So gamma and delta are 3rd and 4th order linear, respectively.
alpha and beta are both second order, but I view them as both non-linear. But I caution on this one, I'm not familiar with quasi-linear classifications.
Alright, the second part is a lot easier than I was trying to make it. You find the discriminant of each equation. If its zero, it's parabolic. If it's positive, its hyperbolic, negative, it's elliptic.
I'm sure you're familiar with characteristic equations, so I'll skip that one.
The last problem is simply doing what you know to do, and using the initial/boundary conditions to solve the PDE. Solve the general form as much as possible, finding the general and particular solutions, then solve for the coefficients if needed.
Any problems with solving the PDE?
Well, I have to go. You seem to either be busily working or have stepped away. I hope I helped in some way. Good luck!