"I have the men's books."
Translation:Ho i libri degli uomini.
G like "dissolves" (assimilates) here and not actually pronounced as a standalone sound. You should say something like "deyyi", but it is better if you listen its pronounciation in whole sentences. It may slightly change near stressed syllables and sound closer to "dei" when the neighboring words unstressed.
Yes, Indeed. 'I have the books of the men' is how Italian forms a possessive sentence: 'possessed thing' di 'owner'. La casa di Mario -> 'Mario's house'. Il libro di Giulia -> 'Giulia's book'.
Incidentally, you need to use the article with possessive adjectives. La sua casa -> 'his/her house'. Il suo libro -> 'his/her book'
We would need to know more details, such as exactly how the question was presented to you and exactly how you answered. It wasn't including "io" that got you marked wrong. You either had a typo or error or extra space somewhere, or you didn't select all of the correct multiple choice options, or you dragged from the word bank instead of tapping.
The extra space issue and the dragging issue are bugs, not valid reasons to mark you wrong, so next time that happens (after you triple-check to make sure you had no typos or errors) take a screen shot and fill out a bug report.