"Yarns", definitely, but I can talk about a single thread or multiple threads without it meaning "a kind of thread", as in "my shirt has a thread loose", or "tie the two threads together" (by comparison, to get the same meaning I'd have to say "tie the two pieces of yarn together").
Is a reel some unit of cotton threads? I speak American English, but I have never heard that term. I would expect cotton thread to come in spools (even large industrial ones) or maybe skeins, if there were no object the thread were wound around. It's not something most Americans come across, though, in any case.