To both: Yes, completely. The K-symbol is a special element, and an arbitrary element is a combination of such elements, with coefficents with q-polynomials. The latter you specialize, but you cannot specialize any q "inside" the K-symbol, in principle.

I formulated more carefully, thanks for the comments.

Thank you for the comment...! Indeed, using the keyword "epimorphism" I found statements that if the epimorhpism F is finite, then it is surjective, and if the epimorphism F is flat, then it is surjective. If you have a book recommendation for these or similar facts, please let me know.