"Do you like this train station?"
Translation:Mögt ihr diesen Bahnhof?
Bahnhof is a masculine noun (because it's based off the root word Hof, which is also masculine). Therefore you should memorise der Bahnhof ("the train station").
If you instead want to say "this train station" (i.e. to be more specific) you use dieser Bahnhof. The general pattern to unambiguously say "this [something]" instead of "the [something]" is to use dies- with the same ending as you would need for der/die/das/etc.
This general pattern also applies to how the article gets modified for the different cases. If you've read Duolingo's lesson tips you'll know that masculine nouns in accusative case convert der into den. Hence, dieser also gets converted into diesen.
I'm a German native speaker, maybe I can help out. Nope, that's not correct, because it depends on the case! 'Magst du diesen Bahnhof?' is the only valid answer. The reason is that the train station is the passive object in this sentence. If you said: 'Dieser Bahnhof ist schoen.' the train station would be the active subject of the sentence, which allows you to use DIESER. I know German is not easy, but I feel like you're getting the hang of it :)
Both of them are absolutely correct, and the reason is: 'You' can be translated in various ways, it can be 1) 2nd person sgl.: DU 2) 2nd person pl.: IHR, and 3) the more formal 2nd person sgl: SIE (which you would use addressing a superior at work, for example). For this case that means that, 1) Magst du diesen Bahnhof?, 2) Mögt ihr diesen Bahnhof?, and even the formal 3) Moegen Sie diesen Bahnhof? are correct.
According to what I've learned so far, your sentence translates to "have you like this train station". In general, gern(e) translates to "gladly" and is used to say you like to do something i.e. is used with verbs; while mögen is used for objects. However, I'm still a beginner in German as well so I could be wrong, so anyone with a better grasp of the language kindly correct me if I'm wrong.