"I do not want him to worry."
Translation:Mi ne volas, ke li maltrankviliĝu.
It's not that bad if you understand how it's made. trankvilo is calmness or tranquility -igxi is an affix that means sort of "to become" mal- is a prefix that means the opposite so it means "to become the opposite of calm", or to be worried =D
I don't think they're completely interchangeable. "Zorgi" can mean "to worry" but it can also mean "to care for" or "to look after" and brings moreso to mind the "worrying" that one does when one cares deeply about another person or thing's wellbeing, happiness, etc. "Maltrankviliĝi" on the other hand means more literally "to become anxious/upset/jumpy/agitated/etc".
So it seems to me that if you said "Mi ne volas ke li zorgu", to me it would sound like you don't want him to give a damn (about whatever it is). But if you said "Mi ne volas ke li maltrankviliĝu", I would interpret that to mean that you didn't want him getting upset or agitated (about whatever).
"Zorgi" means to take care of. Of course this can come close, but not in this negative sentence.
That as the conjunction of an object sub-clause. And that is what you have here! The English solves the problem by using an infinitive, thus you do not need a conjunction here.
I'm not telling him not to worry. I'm saying that I don't want him to worry.
I can only guess that this is the subjunctive use of the -u affix rather than the imperative. But that's a complete guess.
I am at a compete loss about how to tell that this shouldn't just use the infinitive. Does anyone have any handy tips?