AO,
Absolutely a gear can get stuck on landing.
Mud...lots of stuff can happen. Hard mud (nearly frozen clay mud, for example) tends to be fairly slick if still slightly wet. If frozen, it's like landing on tarmac. Nothing much different. If thick mud, one can dig in and flip, damage gear, groundloop, skid, make a sharp turn, or any number of other things.
I have seen airplanes stop very suddenly from an approach speed; so quickly they didn't skid. I attended the crash scene of a M18 Dromader many years ago, when they were still very new in the United States. It was loaded with chemical (600 gallons, an enormous chemical load for ag airplanes back then), and thus had plenty of inertia as it entered the field. One of the main gear struck a fence post hidden in tall cochia bushes that were surrounding the field. I'd have bet anything that the fence post could have been uprooted and tossed, or at least broken. It wasn't. The airplane stopped post-haste, didn't even skid, the hopper shattered and the spar broke, and the pilot drowned in the chemical.
Not much difference between approaching the field to spray, and approaching to land.
I've seen and done some extremely short landings in mud and muddy fields. Muddy enough that one had to carry a LOT of power to taxi.
The nice thing about fiction, however, is that your character doesn't need to be gnats-ass accurate. You can make your protagonist rip off a gear, skid, turn over, or do whatever you like. Research and research, but at the end of the day don't lose sight of the fact that it's still fiction, and you have a lot of lattitude as you write that novel. As always, good luck!