Well, as far as I know the name 'Immortals' was never used by the Persians themselves. These must have been the only pure Persian infantry force as Persians switched to horses once they established their dominance in the Iranian plateau. As it was the personal foot bodyguard of the Great King it is possible to have had a fixed strength (10,000 why not?Not all of them participated in opperations...)

And yes, after the Persian Wars they seem to disappear from the scene until Alexander the Great came into play, and particularly in Gaugamela. There, they appear close to the king. The 'units' 'King's relatives' and 'Apple bearers' most probably refer to this elite bodyguard.





PS; I think the 'apple bearers' appeared at Issus too, not sure though...