Re: a sport vs sport
Sports is one of the few nouns that takes a plural when used as an adjective.
We have a shoe store or even a pie shop, but a sports car, a sports arena, etc. When they get to the segment on the new when they talk about how local teams did, it's called "the sports" and in the newspaper it's "the sports section."
In the US, we would say "I enjoy sports" not "I enjoy sport."
If you asked me to describe American football, I may say "It's a sport sort of like rugby." I would never say "It is sport" or "This is sport."
So my short answer to you is "I wouldn't use 'sport' in the ways you have shown."
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.