Is my regular expression and finite automata diagram for this state table correct?
Your automaton accepts all words of even length, whereas your regular expression accepts all words of positive length. The most common way to express the language of all words of even length as a regular expression is $(\Sigma\Sigma)^*$, where $\Sigma$ is a common shortcut for an arbitrary letter of the alphabet (in your case, $\Sigma$ stands for $a+b$).