I'm no expert on ancient sports, but I can answer some of these questions...

In Ancient Greece, sports were very much the pastime of the leisured elite. Most ordinary people could not afford to take the time out of their day to go to the gymnasium, so most professional athletes will have come from wealthy backgrounds. You would likely get into ahtletics because you are a rich young man who has done various kinds of exercise from an early age, and are seen to have a knack for them.

(Nothing about Greek military practices was in any way "regular". Athletic competitions were sometimes held when armies were in the field, as a form of physical training; however, this was far from a universal practice. I do not know of any examples of athletes who were "discovered" while on campaign.)

Once you decide on a career as an athlete, that will be your life until you are too old to compete. Professional Greek athletes followed a special diet to bulk up, mostly involving excessive amounts of meat. They slept a lot, and exercised the rest of the time. Because of this regime, athletes got used to having more food and rest than ordinary citizens; Greek military thinkers heavily criticised the lifestyle of athletes, who could not handle the hardships and deprivations of military service. Despite their strength and speed, they were considered unfit for war.

You would be able to compete frequently, though. The 3 Games you mention (and of course the Olympic Games) operated on a cycle, but there would be local festivals as well, and games to celebrate special occasions. By Roman times, there was a well-travelled circuit of festivals throughout the Eastern Mediterranean that kept athletes busy for most of the year. Leading athletes would be superstars, famous throughout the Greek world (and later the Roman Empire) - winning at the Olympics was such a big deal that the historian Diodorus frequently records Olympic victors under the relevant year in his universal history of the Greek and Roman world.