Sorry to be that guy, but acksherly in this case gratiā is ablative, not the nominative that you’re trying to force it into. So it means ‘by grace’ (or ‘by way’). And then, as you say, exempli is genitive. Therefore the two-word phrase can literally be translated ‘by way of an example’.

E.g. and i.e. are common. Other examples of latin used in English writing include scil., short for scilicet, or viz., for videlicet, both in English meaning ‘namely’ or ‘to wit’; and sic, which means ‘thus’, used to indicate that any perceived error is in the source material that you’re quoting. That latter is often wielded as quite the slap down!

No. It translates to ‘by way of an example’.