Purchase them with Odin marks from vendor in the Palace (in Chapter 9). Once you put one on a character, XP earned automatically earns on the Legendary.
Hard one to answer, as it depends on characters, builds etc.. Basically though, trash Normal and Heroic versions of quest artifacts, non-advanced versions and pretty much every artifact with an item level below ~30 or so.
Uniques, again - hard to say. Some have benefit, some don't. Probably best to check for a preferred build. Generally though, most of the Any Hero Uniques are crap - but you wouldn't trash them, they're good for Exchange fodder at the Crafter.
Cosmics, eh, I don't keep any, but there are odd characters without QoL or 52 review passes that can have better rolled cosmics than their Uniques, hence BiS, but I wouldn't be keeping any on the off-chance personally.
Uru - I keep two of each, of all types.