I'll describe your mistakes first, but the problem here isn't the mistakes you cite. It's that you don't understand what you're doing with the induction. I'll answer the real problem after commenting on your "mistakes" below.
First mistake: why even say x+y >= 4? Why is that necessary or helpful?
Second mistake: No. The problem is you're claiming a certain property exists wrt "4x+5y" without describing how to arrive at the values of x and y.
Third mistake: This is the same as second mistake.
Final mistake: Your formula isn't correct if it allows negative values.
But those aren't really the mistake here. Here's the problem--you don't prove things by induction by manipulating the same variables over and over. You do it by showing that based on previous cases where a property holds, successive cases must also hold.
I'll give you a hint: you should really be using strong induction here. Try this format:
1) Prove that for n = 12, 13, 14, and 15, P(n) is true.
2) Assume P(n), P(n+1), P(n+2), and P(n+3) hold. Let a = the value of x such that P(n) holds and b = the value of y such that P(n) holds. Can you show that this must imply that P(n+4) holds?
3) If so, by FIP, you have proven P(n) holds for all natural numbers 12 and larger.