Hospital tanks should not really have sand or live rock, but, you can leave the sand but DO remove the rock.
The reason I say leave the sand at this point, is because I am assuming that your QT has not been painted black on the bottom, right? If not painted black [on the outside underneath the tank] then the fish will be stressed by the reflections of a bare-bottom tank. Thus, leave the sand in place but take out the rock. You will need to create comfort zones in place of the live rock. PVC pipes is a good choice for this purpose.
Yes, you will need to move ALL fish exposed into the QT. A 10 gal is rather small. How many fish are we talking about here? If you have copper, begin the threapy immediately, as, with velvet, there is absolutely no time to waste. If you do not have copper on hand, then rush to the LFS immediately and get some.
If you end up with ick instead of velvet, then the copper will also treat the ick...however, for ick, I prefer to recommend hyposalinity.
What makes you think your fish have velvet?