Regular but not normal space

Suppose $\chi$ is a regular space. Suppose $A,B$ are closed disjoint sets. If one of them is a singleton, then the regularity implies that they can be separated by open sets. Otherwise, since the sets are disjoint, then each of them is of size 2.

If this is the case then $A= \chi - B$ is open as a complement of a close set so $A,B$ are open and you can separate A from B, therefore $\chi$ is normal.

No, it is not possible. For example check Munkres's Topology book, p 200:

Theorem 32.1 Every regular space with a countable basis is normal.

In case of finite topological spaces regularity and normality are equivalent.