Why isn't it "estas necesa"? An adverb seems to make it sound like "It's necessarily" instead of "it's necessary".
In sentences without a subject (which in English get the auxiliary subject "it") you use the adverbial form. This is however only explained near the end of the course.
Which lesson is that explained in? I didn't notice it, though I've finished the tree.
Why are these using the imperative form when English and German use the indicatives?
In this case Esperanto behaves like the Romance languages, which use the subjunctive mood. The subjunctive is used in situations of doubt, desire, opinion, and others. The conjugation of the verb in the subjunctive mood is like its conjugation in the imperative mood.
I am under the impression that Esperanto allows both manĝas and manĝu here, as using either doesn't really affect the meaning.
In this way you can use whichever makes most sense to you, no matter what your mother tongue happens to be. (Zamenhof allowed for some "wiggle room" for cases such as these.)
Just be aware that the other way is also considered valid Esperanto. So when you encounter it, accept it as valid (despite the fact that it is not how you would normally say it) and you should do fine.
So either Manĝu doesn't mean "dine" or possibly "sup." Just the basic and somewhat crass "eat." The CEED disagrees on that first one, and suggests acceptance on the second.
I've reported it.