Imaginary numbers are used a lot in engineering principles and such. I did a course on them in my final year of university but a lot of the stuff I covered escapes me now haha. As for negative numbers, think about opposing forces. So you're driving your car and you have a forward driving force and the backwards force is the resistance. This backwards force can be considered as a negative number to represent a force acting in the opposite direction to the driving force - it all depends on how you choose to label your diagram when you do these kinds of calculations.
Feels to me like mathematics is an interpretation of nature. Nature is discovered, but nature is not math. Math is the language we use to understand nature. In that sense it feelss like it's more communication than anything, a worldview or conception. I think there are several ways of understanding nature, math being just one of them.
My response is in red and I agree with what you say in your second paragraph, except for the double s on the word 'feels' :p