You will not want to use dishwashing liquid or a "scouring pad" to wash the bike with. Dishwashing detergent will first, strip your wax. Second it will, over time, dry out any seals, rubbers, or adhesives. The scouring pad on paint is just a bad idea all together. You are putting scratches in the surface of the paint, and then filling them with a silicone to "hide" them. Be better off not putting the scratches in the paint in the first place.

I own a detail shop, and when someone comes to me with faded paint, it takes a light polish or compound to bring it back, and seal with a sealant or wax. Nu-Finish will work fine, but it stains anything rubber, so be careful. I like Mequiars Color-X for people that arent using a several step buffing process. Color-X wont stain trim, has a chemical cleaner to remove oxidation, and durability on a bicycle should be fine. Use microfiber to apply and remove, and your paint should look much better.