Not sure I agree with the whole post in detail. Distinguish "pure algebra" from "applied algebra"; and within "pure algebra" distinguish "structural" issues from "combinatorial" ones such as the Burnside problem. Remembering that "abstract algebra" is the modern term for what used to be called "modern algebra", we should probably drop the "abstract" to get a more reasonable view (the scope of "old" or 19th century algebra being that of Chrystal's Algebra say, some would now count as other branches of mathematics, such as numerical methods).
So which questions are worth studying? Not just one kind, surely. Algebraic geometry, algebraic topology, algebraic number theory all do ask serious and interesting algebraic questions. See for example the Golod–Shafarevich theorem (http://en.wikipedia.org/wiki/Golod-Shafarevich_theorem) which is pure algebra to start with. Parts of algebra come across as "general" compared to mathematics as a whole, but this is somewhat subjective criterion these days. There are both general-structural and general-combinatorial parts of algebra. There do need to be some criteria operating in, say, infinite group theory and infinite-dimensional Lie algebra theory. Generality in the sense of category theory is rather 1960s in feel; derived categories are "abstract" but I wonder who these days would argue that they are too "general"? I suppose the general module over the general ring still looks troublesome as a setting for research.
Well, I think "follow the masters" is probably the best advice,