From my massacred reading of the Oxford English Dictionary (aka, people who are working on PhDs in Classics would be much more helpful) -
The origin of a musician's "gig" is unknown, but dates back to c. 1920 in print.
Ovation, in the context of applause, first appeared in print in the 1830s. It largely came straight over from Latin, and in Roman times meant an entry that was a little below a triumphal one. The Latin ovant shares a root with the classical Greek ovation. Appparently ovum, and ovulation, have that all important u that leads back to the Latin word for egg. So ovation and ovulation are totally, word wise, unrelated. And biologically pretty unrelated as well...
School of Music people are a riot, for the record. Yay for cake binges!