Actually, "The Restaurant Menu" should be "an biachlár bialainne" (The menu of a restaurant)
hata + fear = hata fir
A hat of a man
hata + an fear = hata (de chuid) an fhir
A hat of the man
an hata + fear = an hata fir
The hat of a man
an hata + an fear = hata an fhir
The hat of the man
There are no options available to choose. (Reported May 2019) I can find no way past this - no option to 'Use keyboard", "Refresh" just brings me back to the beginning and the question then repeats at some stage. 'Skip' moves on to the next question but then this one comes up again later and still has no options. Any suggestions?
Given that this is a Sentence Discussion, and that the only people reading Sentence Discussions are other people who are trying to learn Irish, I can suggest that you post your question in the Troubleshooting forum, or take a screenshot that demonstrates the problem, and submit a bug report with the screenshot. Adding details like whether you're using the web interface or one of the various different mobile apps might help too.
I'm afraid your understanding of how the possessive apostrophe works needs some work.
The restaurant's menu - the menu of the restaurant (one restaurant)
The restaurants' menu - the menu of the restaurants (more than one restaurant).
Yes, you can also say "The restaurant's closed", but "closed" isn't a noun, so there is no confusion - that's not a possessive apostrophe, because a restaurant can't own a "closed".