दुखी / ऊदास ... do they mean the same kind of sad, and one is from Persian, the other from Sanskrit? Or, do they actually have slight semantic differences?
Wiktionary says both mean sad/depressed, but for दुखी adds sorrowful/grieving/distressed/pained, and for उदास (not ऊ) also gives gloomy/melancholic & indifferent/stoic/apethetic.
So it seems दुखी is used for the more severe/reasoned sadness.
Is the usage of हो here a general principle of the form, or does is mean that "tum" is used for adressing the person?
Because you don't usually explicitly say तू/तुम/आप when ordering someone, there are only really two cases here:
Ordering someone in general:
दूखी मत हो!
Ordering someone you respect (which doesn't really happen very often):
दूखी मत होइये!
It's the imperative verb here - since दुखी is an adjective - so the sentence is literally 'be not sad'.
It does also mean that either तुम or तू is the implicit pronoun, since the conjugation is different for the 3rd person plural (आप).