(1) ends in ㄴ다 so it's in the plain or diary form sentence ending. This form is usually deemed 'expressionless' and therefore used in writing. Another use of -ㄴ/는다 is when exclaiming something to oneself. It can emphasize your first impression of something like "맛있겠다!" or "귀엽다!"
(2) adding -기 to the end of verbs is another way to turn them into nouns but they are often used for fixed nouns such as 웃다 and 웃음 ("to laugh" and the noun form which is translated to "smile") or 걷다 and 걸음 ("walk" and the noun form "step")
(3) In my opinion I think this is the best translation, however I think it should be conjugated as "이기는지 몰라요" since the thing you're not knowing how to do is in the present tense.
-는지 adds a sense of uncertainty, and you put it on things that you don't know or someone else might not be sure of, therefore you will often see the structure "XX은/는지 몰라요"
Ex: I don't know how to cook. = 어떻게 요리하는지 몰라요.
I don't where the store is. = 가게는 어디 있는지 몰라요.