Q3: one of the rules for this is to be far from edge. Does it imply to have infinite length parallel plates? What practical use case would this have if we only have the system defined for infinite plate?
Yes, the perfectly uniform electric field is only going to happen if the parallel plates are of infinite area, infinite in length in both directions. We are essentially saying that we are approximating a finite parallel plate capacitor as a section of such an infinite parallel plate capacitor. It is tolerable as long as the capacitor plates are such that the edges make up a tiny portion compared to the bulk.
Q1: Why is the electric field uniform between parallel plates and is it only uniform between? and not outside?
Because of the approximation of using the infinite plates as the idea for the finite ones. It is fake, but it is likely to be tolerably good enough.
In reality, the electric field can never have that sudden stop, and so some of it must spill outwards.
Q2: Why uniform? Which rule exactly points/proves this out? Putting test charge in any place would feel the same force? how? What If I put it positive test charge closer to negative plate - shouldn't the force be stronger? The rule of uniform charge distribution doesn't prove it to me.
Either you do the integral yourself, fully, or you learn more maths to bypass the need for that integration. There is no point asking other people to prove something to you that is going to be impossible to read to an understanding.
The charge will not feel any more forces anywhere in this region, up to this level of approximation.