Homebrew equipment suppliers sell the standard vinyl hose in different diameters. The one that works with my bottling wand & bottling bucket spigot is the same diameter that came with my original kit. I bought a replacement a while back, I think it's 5/16", but your equipment may vary.
I never cool my priming solution. It goes right into the bucket, boiling hot, just after I start racking into the bottling bucket. As the previous post says, 2 cups isn't enough to kill more than a tiny fraction of the yeast involved before the temperature of that 2 cups is brought down close to the ambient.
“Malt does more than Milton can / To justify God’s ways to man”
-A. E. Housman (1859–1936). A Shropshire Lad , 1896.