No. The Sorgenfrey line (the real line with topology generated by half-open intervals (closed to the left and open to the right, say)) is a standard example: it is separable, but is not second countable, hence is not metrizable.

No. Take $[0,1]\times\{0,1\}$ with the lexicographic order. This gives a counterexample --- it is separable (for example $\mathbb{Q}\times\{1\}$ is a countable dense set), yet it is not metrizable. One way to see this is to notice that the subspace $[0,1]\times\{1\}$ (homeomorphic to the Sorgenfrey line) is not second-countable, hence not metrizable. The counter-example can also be viewed as an example of an Alexandroff "double-point" construction, which is an example of the general construction of "(special) resolution" (which is a nice technique for generating counterexamples).

(Edited to incorporate comments --- original answer was incorrect, citing Sorgenfrey line as a counterexample.)