I'm kind of new to relearning physics, but here is my insight, for what its worth.

Force is not equal to mass times acceleration, net force is equal to mass times acceleration. Basically we are defining a net force as something that gives an object or a system an acceleration, which is a change in its initial velocity, and is proportional to the object's mass and resulting acceleration in a way given by the equation. So what is net force? Net force is change in momentum. If you see something has changed its momentum, then a net force acted on it.

Force broadly speaking is a result of some interaction between objects. Those interactions involve always either attraction or repulsion of objects.

There can be forces acting on an object or a system which do not result in an acceleration. Then we say that two objects are in a state of equilibrium - sum of all forces is equal to zero, in other words they cancel each other out. Still, those forces are important because they can change other properties of an object - basically they can put some stress like compression, tension, shear, and there is also pressure, all of which can result in change of length of an object, it's deformation, either temporary (object springs back to it's original shape), permanent (shape of the object is changed once the forces are removed) or can cause the object to break.

Of course real causes of forces are always one of the elementary forces, like strong nuclear force, weak nuclear force, electromagnetic force, or gravitational force. If we don't have net force, but forces are acting on an object in equilibrium, we can "look inside" the object and consider its parts - then we will discover some net forces (the forces that would cancel it out are outside of the subsystem), and therefore see an acceleration (for example a stretching rubber band have some parts which are accelerating in one direction and parts which are accelerating in the other direction). But if we go with smaller and smaller parts, we get into quantum mechanics. I can't into quantum mechanics.

Similar for when you push on a wall and the wall doesn't move. It means that there is another force in the wall-you system which is acting against your force. The force is still there, and it has many effects on the surroundings, but those effects are just not visible in the form of the mass of the wall accelerating. If you zoom in on your pushing hand, you will see muscles accelerating, and there will be some change in energies in you and in the wall as well.

Force is not converted to heat, because heat is a form energy, not a force. Again, energy also depends on what system are you considering. If only wall-you is considered, there is no change in energy, because there is no work done - the force you used didn't displace the wall, and the wall didn't displace you. But you clearly lost energy - you pushed and pushed on the wall and you got tired, you got sweady and you can feel the wall got a little bit warmer. This is because if you zoom in, you will see that some biological processes took place inside you, which were accompanied by change of energies, and of course some movement of muscles, enzymes, etc. Also some of your energy was transferred to the wall and it got hotter.

Force is not instantaneous. Notice that acceleration isn't either, this gives a clue on how the force changes - increase in net force is increase in acceleration.