1) That formula is only an approximations, it only works with things that aren't too fast, or too small (or both).

2) Cause it doesn't. Not much we can say, Nature behaves like this: we find that there's a linear relation between the force applied on a body and its acceleration, we call the proportionality constant "mass" and, by definition, the mass define the inertia of an object (that is, its resistance to acceleration when pushed by a force).

3) There are some certain scenario in physics in which mass isn't constant or it's not a scalar and it can behave weirdly. In particular in certain crystals, electrons and holes move AS IF they had an "effective mass" that is different from their usual mass. This effective mass can be negative (the electron moves backward when pushed forward), can change with the parameter of the crystal and can be a tensor (a matrix, so that the effect of the force depends on the direction of application), thus creating some really strange phenomena.

4) In relativity some people will say that mass changes with speed and so we can somehow say that m depends on the force, but the "mass changes with speed" thing is really old and not used anymore, it's not a really useful concept and it does create a lot of confusion with little gain. Nowdays with mass in relativity people mean "rest mass" and this is constant (in closed systems) and invariant (so all observers agree on the value of the rest mass).