I am not sure about the cycling ammonia reacts differently to water with a ph under 7 its less toxic, my guess is that cycling with the water your fish will be living in would be safest, though I don’t know for sure. Yeah using filter media from a cycled would speed up the process.
The rummy and the cardinals would be very happy in discus water, its better for them and they wont bother the discus. I don’t know about the cats as I haven’t kept them. Clown loaches and angels I have kept with discus without any problem. I have read that angels carry similar diseases to discus and are more immune to them than the discus, so its often said that angels are a big no with discus. This was not my experience.
The fish you have now would need to be gradually acclimatised to the discus water before they could go in. You would also need to check with the discus seller what water they are in so that you can match that as closely as possible when they arrive.
Yeah impatience is an issue, but you will thank yourself if you do it properly from the start. Happy discus are a great joy, unhappy ones won’t last long.
Good luck with them
Liam