Do people really say this in Turkish like we do in English or is there a more correct and less literal way of translating it?
"birinci" is normally used when you are talking about the physical position of something as compared to something else or for counting things. (Like the 1st floor).
In terms of just something that happened before other things, you would use "ilk." It is a bit more temporal instead of spacial in meaning, I suppose. :)
Aşk usually mean something deeper, passionate and stronger. Sevgi is more like mother and child or to the nature... But they both mean love.
It's 'sight + at'.
görüş = view, sight, opinion. (https://en.wiktionary.org/wiki/g%C3%B6r%C3%BC%C5%9F)
The -te is not dative, but rather the locative ending, the e harmonising with the ü of görüş and the d mutating into a t due to the unvoiced final consonant, ş (http://www.turkishlanguage.co.uk/conmut.htm).
Would "İlk görüşteki aşk" make sense too, or is "İlk görüşteki aşk" just a set phrase?