For (1), if the topological space X is countable, then a countable basis can cover X. Thus, X is second countable (If X is second countable, then it is automatically first countable). By Urysohn's metrization theorem, every second countable regular Hausdorff space is metrizable (I denoted as a regular Hausdorff space. Otherwise, you can also use the fact that every regular Lindelof space is normal. This issue is discussed here).
Thus, (1) is correct.
For (2), the counterexample can be a lower limit topology on R, denoted as . has a countable local basis at each point , where is a countable local basis at x. It is separable since the rational numbers are dense in .
has no countable basis. Let B be a basis for . For each x in , we choose a basis element of B such that . If , then , since and . Thus, B must be uncountable. It follows that is not second countable.
Thus, (2) is not correct.
Some additional remarks: The regularity hypothesis in the Urysohn metrization theorem is a necessary one, but the second countable condition is not. You can use other conditions to check metrizability (link).