Does dark red exist?

Dark red sounds ok. As a wise old woman once told me, if it sounds right it probably is right. It means you've heard it said enough times to have sunk somewhere into your brain.

You might wish to say deep red as well.

There are of course other adjectives which describe shades of red, like scarlet or crimson, though those are probably brighter than what you mean here. For a really dark red, I'd say ruby, burgundy or bordeaux.
Have a look at http://en.wikipedia.org/wiki/Category:Shades_of_red
though I wouldn't bother teaching them all to your students. Dark red sounds fine to me.

Maybe maroon or burgandy, but along with everyone else, dark red sounds fine to me.