Is there any evidence to suggest that there were ways for ancient athletes to get unfair leg-ups over the competition? Were there referees/judges or what level of rules can we likely point to?
If there were rules and one was caught cheating, what kind of punishment could be expected?
So. Much. Cheating.
From our absolute earliest literary sources that discuss games, like Homer's Iliad, game participants argue the outcomes of competitions on the grounds of fairness. In Book 23, the Greeks hold a chariot race, refereed by Achilles, with prizes for the top three finishers out of five. Menelaus, who finishes third behind Antilochus, accuses Antilochus of cheating to take the prize, and Achilles wants to award second place to Eumeles--who didn't even finish the race because the goddess Athena intervened and smashed his chariot--for his superior skill. In the end, Menelaus agrees to settle for third place and Achilles is told to give Eumeles a reward out of his own tent if he wishes, and to respect the order in which the athletes finished. So cheating accusations are about as old as funeral games.
They are also built into the ancient world's most famous sporting event. One of the founding myths of the Olympic games was the story of Pelops' chariot race against King Oenomaus to earn the right to marry the king's daughter Hippodamia--a race Pelops won by bribing Oenomaus' charioteer to replace the linchpins of the king's chariot with wax so that they dissolved at high speed, resulting in the king being dragged to his death. However, because the panhellenic games like those at Olympia and Delphi were fundamentally religious events, athletic prowess was seen as a sort of offering in honor of the gods and cheating was seen as very disrespectful towards the gods.
Olympia gives us a lot of information about the rules and organization of sports competitions and about how cheaters were punished. The Olympic games in particular were organized by local officials from the nearby city of Elis, chosen by lot, called the Hellanodikai. They were in charge of preparing the site for athletes and spectators, organizing recruitment and getting the word out across Greece that it was time, choosing competitors for the limited slots out of those who wished to compete, and for judging the competitions.
A bronze tablet from the end of the 6th century BCE (the unfortunately partial SEG 48.541) found at the site lists some rules like "The wrestler is neither to break a finger..." and "...drachmas are to be paid if he does an injury or..." But then again, Pausanias (Description of Greece, ca. 170 CE) tells the story of a wrestler named Leontiskos who won twice at the Olympic games who "did not know how to throw his opponents and thus beat them by bending their fingers." So it's often hard to tell to what degree some of these rules were enforced.
Cheating was definitely frowned upon in most cases (outside the realm of mythological cleverness). Lucan (On Slander, ca. 170 CE) writes that "a good runner thinks only of what is in front of him...but the bad runner and worthless competitor has no hope in speed but only in tricks and he thinks only of how he might hold up or trip the runner."
Besides sabotaging equipment or being unsportsmanlike towards your competition, you might try to bribe officials or your opponents to throw the match. Outside the stadium of Olympia, there are sixteen (remaining) statue bases that once supported bronze images of Zeus called the Zanes. These were erected using the funds of fines levied on cheaters at the Olympic games, with inscriptions that monumentalized and shamed them for their actions. Eupolos of Thessaly had to pay for the first, for bribing three of his competitors in boxing. The inscriptions also mention physical punishments levied upon cheaters, and bans from future competitions. Every athlete, already having had to take an oath not to cheat at the games before Zeus, had to walk past these statues as a warning as they entered the stadium grounds. Public shame was a big deal in ancient Greece, but not enough of a deterrent that there aren't many examples preserved of people who tried to cheat.
Stephen Miller's Arete is a sourcebook that collects together a boatload of primary sources regarding ancient Greek athletics--all of my translations here are quoted from his work. You can also find further information on cheating in David Potter's The Victor's Crown, a more general history of Greek sports and their importance in the civic and religious fabric of the culture.