Both hit and struck work well. Hit is slightly less formal: I might be inclined to tell my wife that a car hit another car, but I would tell the police the car struck the other car.
Beat (no "-ed") usually suggests a series of strikes. For example, "He beat his head against the wall" or would mean he hit the wall with his head several times, but the exact number is not important. Likewise, to say "The wife beat her cheating husband" means she hit, kicked, or threw things at him for a few minutes.
There are some other small changes (or "tweaks") you might make:
During the competition, my rival hit my leg hard with his stick.
During the competition, my rival struck my leg with his stick.
Adding to, like sentence 1 of your example, is unusual for violent verbs like "hit", "beat", or "struck." In fact, adding "to" can suggest gentleness or exactness: you might say, "The mother pressed her nose to her daughter's head," or "The electrician touched sensor to the live wire."