What's the history of Spurs and Stetsons?

So a couple of things. First, there are two ways to earn your spurs. The first is the spur ride. The idea of the spur ride within the US Army goes back to the origins of mounted troops, including the Dragoons, Cavalry, and Mounted Riflemen. Simply put, it is impractical and unwise to issue the means to goad a horse into action when the soldier isn't able to control the animal without them, because digging your heels into a horse's flank isn't as likely to cause the animal to go charging solo across the battlefield as it is if you dig spurs into them. Now that we don't (commonly) use horses as a means of conveyence, spurs and spur rides are more symbolic, with the latter involving a 3-5 day FTX in which the prospect is tested by "troopers" who have earned their spurs in various and assundry Warrior Tasks, such as marksmanship, land navigation, physical fitness, basically everything you learn in Basic/AIT + the squad level tasks for your unit. At the end of the ride, those who are successful in passing the test are awarded silver spurs. Those who complete a combat deployment with a Cavalry unit receive gold spurs. (Personal note: When an attempt at shaming me into participating in a spur ride was made after my first deployment with 3d ACR, I pointed out that my spur ride lasted a year and included live fire, so I didn't see a point in going on a 3 day hazing trip. I had a similar answer when encouraged to test for EFMB while wearng a CMB.)

The Stetson as headgear within the cavalry is more of a symbol of heritage, setting apart cav troopers (not just 19 series, but all members of a cav unit) from their armor and infantry contemporaries. A couple of notes on that. The cords are denoted into commissioned, warrant, and enlisted. The latter are normally branch colored, so Cav & Armor troopers wear cav yellow, while infantry wears powder blue. Mine was and is medical maroon. The insignia is generically the unit brass to the fullest extent feasible. Some alteration of the basics are tolerated...for example, it is uncommon to see an air cav pilot whose wings aren't affixed with the shield of the wings centered at the point where the sabers cross. Squadron/Regimental numeration is common, occassionally someone will go so far as to have a Troop designator on them as well. In my personal case, I wore my 3d Cavalry sabers with an engineer castle centered on them, which denoted me as being assigned to the Regimental Engineer Company (although now there's Pioneer Squadron.) One final note is the combat knot. The standard rope has two acorns with about 4" of cord back to the basic knot. A combat knot, which denotes combat service (unit immaterial IIRC) involves tying a single half knot with those, shortening up the cords. This is significant since normally they'll hang over the brim, making it easy to visually recognize a veteran trooper. The Stetson is considered optional headgear for anyone assigned to a Cavalry Unit during approved functions.

In 3d Cavalry, at least, authorization for Spurs was accompanied by a 8"x14" certificate suitable for framing inducting the recipient into the "Order of the Stetson & Spurs." Much like St. Barbara and other branch specific awards, it is purely honorary, and means little outside of the Cav...unlike the others, however, it is usually available to ANYONE assigned to a Cavalry Unit, so the Postal Detachment assigned to the Personnel Battalion under the Operational Control of 1st Cavalry Division is eligible for that recognition.

On a final note, there is some disparity in the design of both Stetson & Spurs. Although most officers and senior enlisted personnel stick to the same style of hat, as long as it is solid black in color and maintained well, the design is not sacrosanct. The same with spurs. Although the McClellan style spur is considered to be the standard, the acceptable designs are generally more flexible, within reason.
Although squatting with spurs can be uncomfortable, my advice would be not to put them on until you are at your final destination where wear is expected. Walking down stairs can be near fatal, and driving with them can be problematic as well.

Gold spurs are awarded for deploying with the cavalry. Silver spurs are awarded after successful completion of a spur ride. Rowls can be worn to denote multiple awards of spurs (i.e. gold spurs with a gold rowl means multiple deployments with the cav.)

Goes back to the old Horse Cavalry days. And yes, there are different spurs. You have your silver spurs that you earn on stateside Spur Rides. You also have your Gold Spurs that you get when you do a Spur Ride in a combat zone. And while I know the following link is Wikipedia, it still gives some good info.

https://en.wikipedia.org/wiki/Order_of_the_Spur