Why is it ho and not reho, from rehana
दुखी मत हो! - Don't be sad - from होना - 'to be'
दुखी मत रहो! - Don't stay sad -from रहना - 'to stay'
I believe because you are telling them to be something not remain or stay something. Also, my understanding is that rehana is like a helping verb and isn't the only verb in a sentence often.
Why is मत after दुखी? and why use हो?
The negation मत should be adjacent to the verb हो. दुखी is an adjective.