Squad - check military alert screen and make sure there is an A after the squad. Have you set up a scehdule? They will train more frequently with
certain tweaks to their schedule. Artifact pick - it's worth a happy thought for one lucky miner. Don't try to use it as a military weapon though.
Vomit/Blood cleanup - Good luck, it's just a super low priority. The technique I've used is to set a meeting zone on top of it, that seems to get them to clean it. I don't even bother now - check out the DFHack "clean" or "spotclean" command for mapwide and tile cleanup. Best way to avoid vomit is to ensure that your dwarves don't get cave adaption. This is the main reason I build a surface fort of some sort; my military trains outside to avoid cave adaptation, and I set up a meeting zone in a nice statue garden and that covers most of the civilians*. Your problem will always be the super-busy underground dwarves that don't have time to dawdle on the surface - miners, engravers, etc.
Animal fat comes from certain animals. I always embark with tons of turkeys because they are cheap and can be turned into meat, fat, bone, and a hide. Trade for larger animals from the caravans and you'll get some when you butcher them. The kitchen will automatically pick up a 'render fat' job and turn it into tallow by default. Problem: by default, tallow is used in cooking food - go to Z menu, Kitchen, find the tallow, and toggle 'Cook' to red, and now you'll stockpile tallow instead of eating it. From there you need to make an Ashery and burn some wood into Ash, then turn the Ash into Lye (need barrels). once you have Tallow and Lye the Soapery comes into play.
* others channel down into the earth so that underground spaces are opened to the Outdoors/Light, then cover it up again with constructions. Once a space gets Light it's permanent even if covered by construction, and that space will not cause cave adaptation, so people make meeting halls and taverns and what not under it. The problem is you need 'Outside' to
counteract cave adaptation, and this space is Inside. So the dwarves will still slowly cave adapt from time spent in tunnels, bedrooms, etc unless you find ways to send them Outdoors.