I’ve hunted for pheasants and quail loads of times. I use a 12 gauge with a pheasant-specific load. It doesn’t overly-damage the bird. Yeah, you could get away with a 20 gauge if that’s what you already have available.
Shot - Lot’s of it goes right through the bird. You’ll also find some during the cleaning process, but it’s nothing you need tweezers for. Shot also gets trapped in the body cavity, so you’ll get rid of that when you gut/clean your bird. For a few more dollars, you can buy steel shot instead of lead. Some places mandate non-lead shot. A lot of waterfowl hunting is now non-lead only.
Feathers - The wax dipping has to happen before you’ve gutted/cleaned the bird. I’ve never bothered, but I understand that the point is to pluck the feathers and leave the skin. I skin my birds - so the feathers come right off with it. You can pluck the long tail feathers for decoration, too. I use strips of bacon to take the place of skin in a lot of my pheasant recipes. Speaking of which:
This brined roasted pheasant is the best of many different approaches I’ve taken towards the bird, and it’s a good introduction to the wonderful world of brining. Just make sure you don’t use an aluminum pot.
http://www.recipezaar.com/Brined-Roast-Pheasant-Outdoor-Wisconsin-Style-107369
The hunt itself:
I’ve hunted both with and without dogs. Hunting with some properly trained dogs is fun and easy. I walk a lot of grass and cornfields in general. Grassy waterways on farms are a big hotspot. Fence rows used to be back in the day, but a lot of the overgrown fence rows are gone now. Back to dogs, they sniff out in tight twenty yard circles in front of your party. Some dogs only point, then a party member often must flush the bird. I don’t care for pointers because it takes a lot of surprise out of the hunt. It feels more like a bird shoot. Labrador Retrievers almost always flush the bird. I love that because you’ve got to stay alert for the whole hunt and quickly track, aim, and fire. Pointers make the hunt a very leisurely affair.
I’ve hunted plenty of times with no dogs or with an untrained dog too. If you don’t have a dog, you’ll be doing a lot more walking. Flushing a pheasant sometimes only requires that you walk within ten yards of the bird. Other times you’ve got to kick the grass right next to them - and you will rarely spot a bird on the ground in ever moderate grass; their camouflage is tops. If you’re hunting with a rookie dog, fashion a 15-20 yard lead with some good rope. Tie the lead onto your belt. Some people use shock collars to train the dogs to stay close. I’ve always trained my dogs with clipped-off pheasant wings (feathers on) hidden in tall grass. They get the smell of the bird.
One last tip, a regular pair of plier-sized hedge pruners are super handy for cleaning pheasant. I clip the wings at the elbow and the feet and head clip right off too. A knife works, but if you’re cleaning more than one bird, you’ll want to use pruners.