What complexity class decision problems can be solved by only addition, multiplication, division and subtraction?
The question isn't exactly well-defined, but seems like a good/interesting question to me. I believe the answer you might be looking for is the class of Elementary Recursive functions. This class consists of almost exactly what you described: Problems solvable by functions that can be written with only addition, subtraction, and multiplication of integers. (Except, we also require a zero function and a "projection" function that outputs one of the inputs.)

You can get more functions (the Primitive Recursive functions) by additionally allowing functions that call themselves (but are guaranteed to terminate). Finally, you can get the class of all Turing-computable functions by introducing a minimization operator: find the minimum $x$ such that $f(x) = 0$.

Anyway, it turns out that $\mathsf{EXPTIME} \subsetneq \mathsf{ELEMENTARY}$, so it includes many (most?) problems that we'd usually consider interesting.