I have one small question about Core Melisandre's Asshai strength bonus. Her text box reads "While Melisandre is standing, Asshai characters get +1 STR." As an Asshai character, does the strength bonus apply to herself, bringing her strength from 3 up to 4?

Thank you.

As long as she is standing, sure.

Abilities like Mel's will apply to all cards with that trait, including themselves, unless something specifically says otherwise.