I can answer a few of these questions.
There's actually quite a few different trap ingredients. There are 3 different types of bottles, mechanical parts, magical parts, and a part for every element, plus force parts. That means there are 12 new ingredients. Other ingredients you will need however, are soul gems (which can go into a soul gem bag), and dragonshard fragments, which will go in your ingredient bag.
On live servers, I'd recommend only keep parts you plan to make traps from and will get in high quantities, like the mechanical parts.
As far as I know, there are no secret recipes. Basically all the recipes are outlined in the various trapmaking guides. The only trick is that mines will simply use mechanical parts in place of the different bottles. The simpliest recipe is 10 elemental parts and 10 bottles. The most complicated recipe is 100 elemental parts, 200 fragments, a Strong soul gem, and 10 unadorned bottles.
There is some trial and error in the crafting, but since you don't lose ingredients by trying, it's not that punishing.
As for good places to farm parts, redfang is a decent one. I'd also recommend trying Church and the Cult, and Purge the Heritics over in House P. Gywlans is also good if you can sneak to the various traps.