A 'GT' is typically interpreted as a 'Grand Tourer'. Veloce pertaining to speed 'velocity'. In this instance 'fast grand tourer'. With its root in latin gran or the French 'grand' can mean great, large, big, grand (as in the English 'grand day out' and so on. I believe the racing term 'grand prix' is not so much big prize as great prize, it depends on the usage like much misinterpreted language.
I'm happy to be corrected by someone with an actual knowledge of Italian though!!
Spider was a kind of light, horse drawn buggy with no roof. It got it's name from it's physical appearance.. with the wheels stuck out right in the corners on their suspension.
Coupe on the other hand was a short version of a roofed coach.. "coupe" meaning "cut (off)" as in shortened.
Urban legend for the origin of the name "Spider" is much more interesting .. apparently the (Graduate) Spider was going to be called "Speeder" but the "ee" sound was lost in translation.. someone at the factory spelt it using the Italian vowel "i" which is pronounced the same.