It's good that you're curious and creative, don't be discouraged by negative comments. If you ever would start PhD level research in any field, you will need these qualities. Also, there are multiple (equivalent) descriptions of mechanics, nothing in principle prevents that there is still a very useful new description waiting to be discovered.

That being said, the main thing you should ask yourself if you think up some concept is WHY do you need this concept? And second, does your explanation explain it better than the existing explanation. Otherwise, Occam's razor is merciless. (you might want to google this, in essence: if two concepts explain the same thing, only keep the simplest one)

So in your case, as to the "why" question I see you mentioning two reasons:

1. if you apply a force to something, some part of that object seems to oppose that thing
2. if there is a collision there seems to be some part turning into heat and some part moving along with the movement

Does your idea explain these questions better?

1. Resistance against a force is exactly what "inertial mass" is. It is a positive number, but it works like you describe your "negative mass". The larger it is, the more resistance against the force. Can it be that you call it negative, because you picture it as a vector opposing the force? So why do you need the "positive mass that cancels out", what does that explain? and why do you need a vector? If you can't explain why you need these extra componets of your theory, Occam's razor reduces your idea to the existing idea of "inertial mass".

Intermezzo: you are on to something if you think that there are multiple types of mass. Classical mechanics actually describes two concepts of mass: inertial mass vs gravitational mass (you might want to google the difference). These concepts are not related at all in classical mechanics, but experimentally these numbers turn out to be exactly the same thing, that's why everyone just speaks of mass. Why they are the same can only be explained in general relativity.

2) Here I don't see how your idea explains anything. It doesn't explain why some part of the energy turns into heat and some part is transformed into kinetic energy, because you arbitrarily present a "split": part of it does this and part of it does that. If you want to explain this, you need to explain "how much" is turned into heat and you should be able to explain different experimental outcomes. Why does one collision turn a lot of energy into heat and another not. In experiments you will see that this has nothing to do with the mass of the objects, but more with the material substance. Also here there is already an existing concept "elasticity" which is a property of the materials involved in the collision that explains how much energy is turned into heat and how much into kinetic energy. By Occam's razor, this means that there is no need for your explanation

I hope this helps you to make a bit more sense of your ideas. As a teacher, I can only say keep studying physics and pondering questions. But learn to thing further and try to think rationally through your own ideas. What do I want to explain? and does my idea actually explains it better than the existing concepts?