Translation:The next three trains go to the North.
Is it normal to say поезд едет/ездит? Since it's going under its own power, doesn't идёи/ходит work?
Duolingo's sentence is bad Russian. It is understandable, but no native speaker would use "едет/ездит" with trains or buses to describe their proceeding along their routes, only "идёт/ходит".
However, you can use "едет" to describe their current state of motion.
Is there a reason we can't say "The following three trains are going north."?
Because it's a number 'of' trains, after numbers ending in 2 to 4 use singular genitive.
As others have already pointed out, you should use singular genitive here, "пóезда" here. But that said, the plural nominative of "поезд" is atypical; it's "поездá" (pay attention to the stress!) not "поезды". This must be doubly confusing since all other nouns ending with -езд (and sharing the same root with поезд!) have regular plural forms and do not shift the stress: подъезд→подъезды, отъезд→отъезды etc.
Pretty much. It's just that "north" is effectively an adverb in the former expression and a noun in the latter.
'The three next trains' instead of 'the next three trains' is not accepted. Is that not possible in English?
No, it's not possible. And it's not just English -- it's logic. The next train is the first one that comes following this moment. There is no such thing as three of those. On the other hand, the next three trains are the first three trains that come following this moment -- no logical issues here.
Off-topic : be it logical or not, it is the order that is used, for instance, in French: "les trois prochains trains" (the cardinal is always before the adjectives: this is a strict grammatical rule, or so it seems to me). A non native English speaker might not necessarily analyze the cardinal as a regular adjective which can freely be moved around to convey a more precise/more correct meaning.