Are today's physics formulas based around Properties of Multiplication/Division/Addition/Subtraction?

Fair enough question.

Form follows function. In this case, the function is the physics and the form is the shorthand that describes the physics. That shorthand is mathematics.

When you see a product of two factors in a physics equation or formula, it's because the physics, the function, calls for that product.

For example, the physics says if a net force F acts upon a mass m, the mass will accelerate. And in shorthand, in math form, that statement is succinctly written as F/m = a. Or, ta da, F = ma, which is why the force is the product of the two factors, mass and acceleration.

Note in this form... if the mass is not accelerating, meaning a = 0, there is no net force, meaning F = 0. So if acceleration doesn't exist, the way you put it, in the system with mass m, then a force acting on it doesn't exist either.

So the relationships described by formulas and equations only reflect what the physics, the real world, dictates. That's the deeper meaning of those equations.... reality.

Algebraic arithmetic is used in anything that pertains to mathematics. Like a square's not a rectangle but a rectangle is a square, algebra's not trig but trig is algebra, trig isn't calculus but calculus is trig and algebra. Physics uses all of the above. If there's a chair and someone brings up another chair, knowing there's now two chairs isn't "dependent on addition", addition is just one of the fundamental mechanisms in expressing logic. I think you're looking at algebraic arithmetic as a specific technique, as opposed to looking at it as everything.

The functional relationships in typical physics equations come from field theory.