1. A <p B, B takes exponential time. It means A is not more harder than B, So A won't take more time than the exponential. this reduction tells the upper bound, but we can't comment anything on lower bound, in fact A may be solved in O(1) time.
2. A <p B, the best algorithm for A takes exponential time, there is no polynomial time algorithm for B.
yes, this is correct, because if B can be solved in polynomial time, then we can reduce problem A into problem B, and then A can be solved in polynomial time too. But as mentioned, Best algorithm for A takes exponential time.
3. A <p B,
If we have a polynomial time algorithm for A, we must also have a polynomial time algorithm for B.
This is not correct, B even may take exponential time. It only tells that B can not be easier than A, Because if B is easier than A, then We can reduce A to B and can solve A in less than polynomial time.
4. A <p B,
If we don’t know whether there is a polynomial time algorithm for B, there can not be a polynomial time algorithm for A.
This is also false, because if B has polynomial time algorithm then A also has polynomial time algorithm, because we can reduce A to B, and can solve A in polynomial time.
If B doesn't have polynomial time algorithm, then still A can have polynomial time algorithm, because in this case A is not easier than B.
Correct Answer: $B$