I live by Leather New. I've found its the easiest and best for cleaning. I spray it on my rag first and then sudze it up on my tack. If you spray directly onto the tack and don't wipe it with lightning speed, it can run/splatter and won't be applied as evenly. So just spray it first onto your rag/sponge.
Leather New will actually keep your tack fairly well conditioned as well. Unless it's a desperate case (brand new or "been sitting around" tack). I've used Leather CPR with some success and just good ol' Neatsfoot Oil (although, oil will darken your leather).
I've also used the Leather New brand of Conditioner that you linked to and well, wasn't that impressed with it.
I've heard a lot of good things about Leather Therapy products, but haven't tried for myself yet.
Nah, Leather New is a liquid, so you don't need to dampen the cloth.
If it's a REALLY dirty item, be sure to get all the dirt (or anything else that could act like sandpaper) off the tack first. Sometimes I'll use a damp cloth for this first (or dry for a simple dusting), and then a fresh cloth with just the Leather New for the final cleaning.