In this sentence Qo'noS and mon are two nouns and next to each other. In such a case, the first noun describes the second noun. So Qo'noS mon means something like "Kronos's capital" or "the capital of Kronos". To get the meaning you are going for ("the first city of Kronos"), you would need to move Qo'noS to instead be describing the first city: Qo'noS veng wa'DIch.
While the difference is subtle in both English and Klingon, the difference exists in both languages and so we expect you to maintain that difference in both languages.
Yes, but First City is the capital city of Kronos and is a city, not a country. IMO, the translation given by Duo for this sentence should be "First City is the capital of Kronos.". Whether or not you accept " The First City" as correct is another matter. A Klingon would never say "I'm going to The First City", and we should be learning Klingon here.
Well, a Klingon would be speaking Klingon, where there no articles, making the discussion moot :)
That said, I personally treat First City as a proper noun and don’t use an article, but others (including the Memory Alpha and Memory Beta articles, http://memory-alpha.wikia.com/wiki/First_City and http://memory-beta.wikia.com/wiki/First_City ) do use the article, so you may see a mix of styles on this course depending on who added a given sentence — and as jdmcowan wrote, both versions are generally accepted.