I've had elks eat some kind of root that wasn't bog bean (don't remember which one(s)).
Badgers can be trapped, yes. Small deadfalls are too small and bear traps are too large, as are pit traps. Fox pawboards might work occasionally, but it's definitely not the best choice even if it works (I've caught animals that weren't foxes in these traps, but don't remember which species). Badgers are omnivores, and so should go for meat, but I'd try turnips and/or berries. However, badgers seem to be fairly good at avoiding traps.
I use reindeer and elks for dodge and block sparring: Gear up in full protection gear and chase them to exhaustion. Then move around to stand in front of it to get a chance to get it to attack. The ideal is if you can chase them into a single tile dead end where the only way out is through your character. However, expect a training session to take a huge amount of time (and I kill them at the end of if). These animals also have the advantage that they don't aggro dogs, and so don't draw dogs into the conflict.
So big deadfalls (wolf/lynx/glutton trap) are the one to use for badgers? Actually what exactly is the small deadfall even good for? (Big) Foxes, eagle owls, capercaillie, swans and maybe kuikkas? Only one of those you'd specifically want are foxes and they have their dedicated trap already, so small deadfall seems kind of pointless trap. I presume beavers also go to big deadfalls?
When I see badger I immediately mark the spot on the map and then tie my dogs to a tree (always in that order lest I forget to mark it), that way the dogs don't get in the way. I keep my dogs leashed by default anyway, so it's fast for me to do so. Then I run after the badger. Luckily they are among the slowest and least enduring animals, arguably even easier to persistence hunt down than lynxes. The benefit of badgers is that they actually attack the player which results in much more frequent attacks from them and hence faster training. Sometimes they keep aggro on you until they pass out or die, sometimes they forget to attack you at all but taking a punch at their legs is usually enough to remind them of you. Single layer of bear furs is generally sufficient to block their attacks, so you don't need any special armor. Occasionally they'll leap about and bite at your eyes though.