All non fish can stay in the display tank. Corals, anenomes, inverts. Ich won't effect them. Leave them in there for the duration of the fallow period, 76 days. Be sure to feed them now and again.
As for an eel QT, I went the this very same thing with Velvet. I was forced to use copper in my display, as I was at a lack of tanks at the time, so I set up a cheap tank just for the eel.
Materials:
Powerhead
Heater
Light, doesn't have to be fancy
Large, heavy duty Rubbermaid tote
Take some live rock from your display and put it in the bin for biological filtration, run a powerhead pointed towards the surface for gas exchange. Add 1 or 2 pvc pipes big /long enough to use as a hide for the eel. Cut holes in the lid so the light can come through, but leave the edges intact to prevent the sides bulging out. Use screen and weights to stop the eel escaping. Daily 1 gallon water changes and then it's just a waiting game.
Once the 76 day period is done, acclimate a couple black freshwater mollies to salt, QT and treat them for internal parasites, then put them in with the eel for two weeks. If ich is still present, it will show on the mollies and you can treat via Tank Transfer Method.
This has worked so well for me, not only is my eel happy, healthy, and eating regularly, I have coralline algae growing like crazy. Been a week for me and the mollies are showing zero signs of infection.
Things to note.
Beware of cats, now and again I have to swat mine off as it likes to watch.
Plastic and Saltwater don't always mix, and can cause the plastic to become brittle over time. I had my setup crack on me once, so be sure to put it somewhere a leak will be easily noticed.