Not really. While Dublin Zoo (founded in 1831) is indeed located in the Phoenix Park, a large urban park (over three times the size of Berlin's Tiergarten), the term gairdín na n-ainmhithe was not used to refer to the Phoenix Park, only to the Zoo.
Until 1983, Dublin Zoo was the only public zoo in the Republic of Ireland. I'm not aware that there were ever any others (there are now a number of aquariums and "zoos" at destinations like Tayto Park), so there wasn't really any need to differentiate between big and small collections of wild animals. There is also a Zoo in Belfast, opened in 1934.
No. Irish and English both differentiate between the simple present (téann tú - "you go") and the continuous present (tá tú ag dul - "you are going"). The have different forms, and very different meanings.
An dtéann do go dtí an zú? - "Do you go to the zoo?" (ever, on Tuesdays, when the sun shines, etc)
An bhfuil tú ag dul go dtí an zú? - "Are you going to the zoo?" (now)