Doc,
Some thoughts/answers
Do all winches come with a remote? Pretty much any winch worth it's salt will have a wired and/or wireless remote. Boat and logging winches are the types I can think of that have the most limited capabilities.
Mud is quite a varying factor. I'd think a winch a 1-2 times the pulling strength of the vehicle weight is a good start. Most of us use a snatch block for doubling power.
Warn and Ramsey are the big 2 names in winches. If your life and vehicle won't depend on it (e.g. you get stuck near home and can let it sit overnight or tow it out with your truck), you can step down a bit in name brand to save some $.
ATV/UTV winches normally come with 50-100'. Larger models you would put on a truck or jeep come with longer lengths. My route is to:
A) Replace the cable with Amsteel rope. It's as strong as the cable but is very light and doesn't have the issues cable does.
B) Get a few of those 20' long, 2" wide 20K# tow straps. I attach those together for extra length with a clevis or shackle.
OK, all that said, I don't care for an ATV/UTV/truck type winch for the tractor. I prefer something more like a boat winch (an example pictured below).
My reasoning:
a) A mounted winch on a tractor always seems to be in the wrong place when you need it.
b) A mounted winch will get snagged on regularly.
Think of all the crap that will get into it when brush-hogging.
c) A mounted winch's wires are hard to protect and have a tendency of getting torn out the day before you need the winch.
The picture below is an example (but not exactly the same) as what I use.
That 3-bolt plate in the 7 o'clock position in the picture mounts to the bottom of the winch on the side opposite of where the cable is used. That pear shaped oval is meant to fit over a ball on a drawbar.
I find it works pretty nicely on a bucket mounted chain hook as well. Anyway, using that mount (and a chunk of chain if needed to attach to any part of your tractor), you now have a winch you can strap to your tractor to pull it in any direction. An ATV/UTV winch mounted on the front of the tractor sucks if you need to pull backwards.
It's portable. I leave it home until I need it.
Hand-crank capabilities (in case your battery is dead)
If you have a spare battery or are using the truck as your anchor point, you can have the winch away from the tractor (and the mess it's in) and only go to the stuck vehicle with the cable.
They're cheap. About $50 instead of $400. For as infrequently as you need it (I hope), they work fine. Storing it in the shop instead of on the vehicle keeps it in good shape.
Can be used on any vehicle (or elsewhere).
Anyway, that's my opinion and approach. Your mileage (or pulling power) may vary.