I want to comment on your last question: the classes.
The theorems mentioned rightly belong to the area of algebraic topology (the deep ideas behind them are of topological nature and are usually expressed and proven in an algebraic way), but may also appear in the undergraduate course in a seminar on geometry, in a course on complex variables or some type of analysis, especially functional analysis (a big subject which sounded dull to me until I learned about the abovementioned examples). This largely depends on the professor's preferences.
One could also note that you asked a simpler question, just about affine maps. Those will be studied in geometry and that could come with different adjectives, e.g. affine geometry.
The courses on algebraic topology can be taught on different levels, the standard way is to start quite technical. I think you might benefit from building more motivation up front. There are other nice topology facts explained in books or you could ask a question here.