16.3, Due on December 3

Frankly, I've found this whole chapter to be interesting. I love Algebra and this is a type of Abstract Algebra that is interesting. I find it fascinating that is can be used to factor large numbers. I don't understand everything that's going on but the main ideas are really intriguing. One of the things I don't understand is how it's easy to find B!P. That seems like a lot of work.
I also have two lingering questions about elliptic curves:

1. Doesn't a curve technically have an infinite amount of points? When we worked on the elliptic curve Z(E) mod 5 we only came up with 9 points. Are those the only solutions because our solutions must be integers mod 5?
2. When we are dealing with a finite number of points, such as Z mod 5, how can infinity be a point? It's hard for me to grapple that a curve with a finite number of points and finite values for coefficients can have infinity as a point. Am I missing something or is that just a difficult, abstract concept?