1. You are correct.
2. Take any point x in R and fix an open set U containing it. Since the complement of U is finite it contains at least one member of each of K and A. Hence the closure of K is the whole of R, as is the closure of A.
3. Not quite sure what you mean by upper limit topology, but I think it is the topology generated by open sets of the form (a,b]. On this assumption, the closures are, for K, K itself, and for A, (0,1].
4. This gives the set of all non-negative real numbers as the closure of both K and A.