Let me concisely but properly address your questions.
1) Learn about tensor products and multilinear algebra in general. (Since you mention that you read Lang's linear algebra book, there is a more advanced algebra book by Lang, called - guess what - Algebra; it is a standard reference.) These things are totally crucial in e.g. commutative algebra, algebraic geometry, algebraic topology, mathematical physics, etc. They also allow you to give more canonical definitions and viewpoints on classical topics such as the determinant or the trace. Lang's treatment uses modules (over commutative rings), which are the natural generalization of vector spaces, and I strongly recommend you to learn about module theory (also from Lang's Algebra). Then the natural way to proceed is to learn about homological algebra (Lang's Algebra contains an introduction, but there is a lot more to learn). (A professor of mine called homological algebra "smart linear algebra", which is in some sense true.)
2) I actually asked this question myself; a professor of mine responded by saying that "linear algebra is trivial and not a subject of current research", so I can only leave you with this. Note that this is what he says and not what I think; personally I think that there are probably not a lot of people who consider themselves "pure linear algebraists" (in the same way that probably hardly any professional mathematician would described his field of research as "calculus" in the narrow sense), but quite a few who study numerical linear algebra.