“Notes” is short for “lecture notes” and has nothing to do with size.

FOAG is probably the shortest readable introduction to serious algebraic geometry anyone has written. That's the nature of algebraic geometry.

Actually slightly more than that. A college level abstract algebra class is unlikely to cover much of commutative algebra. You'll want to know a fair amount of Module Theory for this book. Reading through the classic Atiyah MacDonald book should be more than enough.

A module is the "same" thing as a vector space (that we all know and love), except the underlying scalars are ring, instead of field (i.e. no division, e.g. integers). So it's like linear algebra when your scalars are stuff like integers or polynomials.