Is it normal for recessed light wiring to be spliced outside the junction box?

That's a blatant fire hazard and violates code on multiple levels—NEC requires all splices to be enclosed in an approved box, all cables to be secured with proper strain relief, and all metal enclosures to be bonded to ground. Whoever installed that did the absolute laziest hack job possible. Leaving open splices resting directly against drywall with an ungrounded metal chassis right next to them is asking for trouble.

To answer your question about box fill: yes, those little Ensenior driver boxes are tight as hell, but almost all of them are rated to accept two 14/2 cables for pass-through daisy-chaining. The trick is keeping your wire management tight. Strip the Romex outer jacket before entering the box, leave about 3 to 4 inches of conductor inside, and don't try to cram oversized wire nuts in there. The easiest way to make everything fit is ditching bulky twist nuts for compact lever nuts or low-profile push connectors. For the knockouts, skip the standard two-screw metallic clamps with locknuts because they eat up way too much interior volume. Grab some 3/8-inch plastic snap-in connectors (like Arlington Black Buttons). They pop into the knockouts from the outside and leave virtually the entire interior cavity open for wire. Knock out two holes, pop the snap connectors in, push both 14/2 runs through, and make your splices inside the metal enclosure. And definitely splice the bare ground wires together with the box's green lead so the case is bonded. Honestly, that flicker you were tracking down was probably caused by one of those loose push-ins hanging in the breeze anyway, so fixing this properly will likely cure the flicker and keep your house safe.

Wait, dumb question maybe since I'm still learning, but do those driver boxes just sit loose right on top of the drywall? Or are you supposed to mount them to a joist? I've been wanting to replace our old recessed cans with these, but I was worried about leaving the box resting on sheetrock. Also, what happens if someone has 12-gauge wire on a 20-amp circuit instead of 14/2? whould two 12-gauge cables even physically fit in there with the connectors, or would you need an entirely seperate external junction box?

Forty years in the trade and these modern tin sardine cans still drive me nuts. Give me an old heavy Halo rough-in housing nailed across the joists any day. If you run into #12 wire like the kid above asked, don't even bother cramming it in that little driver box. I just mount a standard deep 4-square box to the nearest joist, make my feed-thru splices there with room to breathe, and run a single whip over to the fixture. Saves you busting your knuckles trying to fold stiff copper like origami.