Are we living with a sub-optimal mathematics? Is it possible for us to design a better mathematics?

I'm no mathematician, but I believe I have a fair understanding of basic mathematics and I can reason decently. Here's a rant about some things I don't understand.

I don't like the following facts:

1. We have complex numbers, imaginary numbers, and negative numbers
2. Division by zero is not defined
3. Tangent of π/2 is not defined
4. Square root of a negative number is an imaginary number
5. A finite value for π hasn't been reached
6. 0! has arbitrarily been equated to 1
7. ...

Is it possible that fundamental flaws in the basic elements that constitute our mathematics and in the principles that relate these elements to each other are reasons why we have to resort to self-created filler solutions such as complex numbers, imaginary numbers, "not defined", etc.?

I sometimes feel so. I sometimes doubt the need of 'zero'. I sometimes feel that negative numbers should not exist at all (i.e., numbers should only be positive). Not sure, but I sometimes feel that it might be possible to construct a better and error-free mathematics from scratch, in which every operation is defined, and everything is reasonable.