No! Ich can't transfer from one rock or coral to another in the absence of fish, so adding more coral does not restart the clock for the older corals. However, there could be some free-swimming Ich in the water, so I suggest a rinse in between the QT and DT.
AFAIK, Ich cannot attach to living surfaces, so you shouldn't have to worry about anemones (except for any attached rocks). Shrimps & crabs should be safe after they molt.