Setting your 'heat budget,' ie heat range in tank...
IMHO, of all the things you can do to REALLY hasten cycling, this is one. Get your tank to a steady 79 degrees, with a rise and fall of as little as you can---1 degree would be good.

To do this in a way that will be useful in future, set it with everything running as it will run when there are fish/corals. This means your lights on timer, 12 hours of blues/actinics/moonlights, 8 hours of daylight---in my tank, this means...

Difference between a qt and a hospital tank...
There need not be. What describes a hospital tank can be a qt, IF you're willing to do very frequent filter changes, and you can do it as one tank. Follow with me. Understand the trade-offs.

A 'cycled' qt relies on bacteria. You can go as far as sand, rock, etc...or just sponges with bacteria. You put the fish in, you watch fish for four weeks. If you have to treat, you then move the fish to a hospital tank. And if it's a sand-rock...

A few things many people don't realize...
1. bacterial colonization (cycling) goes on for some time after your initial 'ammonia' reaction, in the sense that life goes on colonizing your rock all the way to the core of it. This is why old live rock is tougher than nails---and why just-cycled rock is very fragile. This process (depending on how holey and porous your rock is) can continue for many months. This is one reason why we say go slow at first. Your rock won't be finished colonizing...

Serial killer fish in your tank...what triggers them?
I kind of wince when I hear the phrase 'showpiece fish'. Most tend to be of a pushy species, and if you don't give them enough room...your other fish start disappearing. They don't even have to be that huge a fish---if your tank size makes them feel cramped, they just start adjusting the aquarium population until they feel calmer in the traffic flow. Your other fish, in a healthy tank, just start not reporting for feeding. Sometimes you'll...

When and how to start with corals...
Corals are in a way a responsibility: their collection in the wild likely cannot continue unrestricted: there is already a call for certain ones not to be collected. Communities have taken to aquaculturing them---putting them in lagoons where they can be grown in native waters, rather like farming.

And we can grow them for domestic sale or trade. Certain ones are quite easy to grow and break apart for sale, and if you want to get into this,...