Not all the exercises on Duo are complete sentences.

Usually, phrases are distinguished from sentences here by phrases or individual words not having a full stop (period) at the end, whereas sentences do.

Ym, yn, yng mean exactly the same thing: just like a, an in English, their shape depends on the shape of the following word.

The basic word is yn, but it turns into ym before m (whether the word originally had the m or whether it was a b or p that the yn turned into m or mh) and into yng before ng (as the result of a nasal mutation of yn on g, c into ng, ngh - I don't think any word starts with ng in its unmutated base state).

It's for ease of pronunciation -- pretty much the same principle by which in- in Latin words turns into ing- (pronounced, but not spelled that way) in words such as incorrect and into im- (also in spelling) in impolite, imbibe. Except that Welsh doesn't do the irrelevant, illogical thing and turn yn into yr, yl before r-, l- :)

No, yn/ym/yng means "in"; the "the" is basically yr which sometimes turns into y or 'r.

I don't think "the" ever takes on the forms yn or ym.