For the last question: If you have a particular university/college in mind, contact them to find out whether there are particular mathematical prerequisites that they require to let you enroll in their software engineering or computer science programs. Such requirements vary wildly between institutions.
Beyond formal requirements, the situation is a bit funny. Most of the concrete topics found in an ordinary high school math curriculum is unlikely to have much direct relevance for software development in general. Some ordinary algebra is always good to know, and at some point you'll probably need a passing familiarity with what a logarithm is, but that's not in itself very filling. (For specific fields there are of course further mathematical knowledge that can be handy -- for example, as much vector algebra as you can find if you aim to end up doing graphics).
Nonetheless there's a clear trend that how much secondary-level mathematical education a student has is a quite good predictor of how successful they will be in the software field. Which mathematics that is matters less -- it's more that the same abstract skills are relevant for learning mathematics and for developing software; skills such as being good at handling abstraction, of breaking a problem down into subproblems, to focus on a subproblem without lose track of where it fits into the greater whole, and to do this for multiple nested layers of subproblems, etc. This is often summarized under the heading "mathematical maturity".
It is somewhat debatable whether learning mathematics actively develops these skills or whether it is just that students who already have talent for them naturally gravitate towards learning a lot of math.
There are also less abstract components of "mathematical maturity", such as being comfortable with manipulating symbolic expressions and so forth, which can be a concrete benefit from leaning some mathematics. Again, however, which mathematics one learns is less important here.
What this adds up to is that you probably have the luxury that just following your interests may be as good as any other direction of mathematical study you could take at this point -- unless there are formal requirements to satisfy or you want to target a specific math-intensive area such as computer graphics.