In American English, "for sport" means "for the fun of it" "for amusement", but with a dark connotation. "He tortures animals for sport" "He hunts wild animals for sport" (I want a federal program to arm the animals, so they can shoot back).
I know that UK English at some point in its history also had this meaning for "for sport". Perhaps time has added to or changed this meaning.
In the US, it's definitely "for sports" in the context of this exercise.
No, it is unnatural to say. I couldn't exactly explain why, but in English there are times when you have to suppress this "the" before a noun. Like when you say "Women are always like that", "The women" would sound weird... I don't know why, I only know it out of habit.