1) Do I need to put all of my fish in QT whether they have ich or not, and if so is a 5 gallon large enough for 2 false percs and a royal gramma?
You do need to threat all fish once all fish have been exposed, whether they show symptoms or not. 5 gals is too small. 10 at the very least, but if you can get a 20 gal LONG up and going that will be best. If you don’t have corals/live rock/inverts, you can do hyposalinity right in the display tank. If you haven’t already, take a look at the FAQ TRHEAD’s post on ich.
2) Since I don't already have a QT tank set up, is it ok to use water from the display tank and avoid the cycle?
If the water in your display is good, then definitely use 100% water from the display. If you have a piece of LR you can “sacrifice” to the QT, go ahead and use it. I suggest this because your QT will be uncycled so you will need all the help you can get in kick-starting a new system.
3) What is better treatment, hypo or ich medicine from the LFS?
The only really viable options is hyposalinity and copper. I would definitely not use copper in your situation because you have an uncycled tank. Go with hyposalinity following the procedure outlined in the FAQ Thread.
4) I read in one of the stickies not to have any substrate in the QT, but to have the bottom painted a color so the fish don't get disoriented by too many reflective surfaces. Is having the tank sit on something like black colored construction paper sufficient for this, or do I need to paint the bottom?
Yes, that would be fine. You can add a few cups of substrate from your main tank to the QT in this situation to kick start the QT’s biofilter. If you will not be treating with medication, absorption of the medication by rock or sand is not an issue. Do you have a refractometer or a quality glass hydrometer?