I doubt anything can be said with certainty beyond the brief information on his gravestone in Sicily:
Flamma s[e]c(utor) vix(it) ann(os) XXX / pugna(vi)t XXXIIII vicit XXI / stans VIIII mis(sus) IIII nat(ione) Syrus / hui(c) Delicatus coarmio merenti fecit.
That is; Flamma*, secutor**, lived 30 years, fought 34 times, won 21 times, fought to a draw 9 times, won reprieve 4 times, a Syrian by nationality. Delicatus made this for his deserving comrade-in-arms.
* a scutor or "chaser" was a class of gladiator usually armed with a curved rectangular shield and short sword, and typically paired against a retiarius, a lightly armoured gladiator armed with a trident and net.
** Flamma means "flame" in Latin and is surely just a 'stage' name rather than any sort of family name.
Anything more can only be conjecture.