I only know enough about 3-gun to slightly appreciate what you're saying. This is something I plan on getting into more this year.
What other categories exists in 3-gun? What if I learned and competed with a .226, 9mm, and 12 ga? How many magazines do you need to carry with a .45 if you're only allowed a max of 10 per? What is the typical magazine for a rifle (that's less than .308)?
Great questions. They change the division names about every couple of years, but there are basically 3 other main divisions now, called Factory, Practical, and Unlimited. All are 9mm minimum handgun, .223 rifle minimum, and 20 gauge shotgun minimum. Most run AR-15s of various varieties and expense, 12 gauge autos, with lots of Mossberg 930s, Benelli M2s, and Stoeger M3000s, with appropriate mods for "twins" loading (which is the new hotness) and mag tube extensions appropriate for the division capacity limit. Unlimited allows speedloaders or box-fed magazine shotguns, so you see some Saigas and Veprs along with tarted-up Benellis or Remington Versamaxes with speedloader chutes and optics.
You can google 3 gun nation divisions and read all the rules (assuming your club adheres to 3GN rules, which most are conforming to now). But basically Factory is a 15-round magazine limit in your handgun with open sights, iron sights or a single non-magnified optic and 30-round magazine for rifle, and shotgun can HOLD no more than 9 rounds total. Practical has a 141-mm limit on pistol magazine length but no capacity restriction (open sights still and no porting or compensators), no magazine restriction on rifle and you can run a magnified optic and offset irons if you like, and shotgun can hold more than 8 in the tube (though you can't start the stage with more than 8 in the tube... you can load more after the buzzer goes off). Unlimited is pretty much anything goes, and most run "race guns": ported or compensated, optic-equipped handguns with huge 25+ round stick mags, lightened, compensated, tarted-up rifles with bipods if desired and definitely a good magnified optic and maybe an offset micro red dot, and ported or comped, optic-equipped, speedloader-friendly shotguns. Rifle comps are allowed in all divisions provided they are less than 1" in diameter and less than 3" long.
Get out there... it's a lot of fun.
What if I learned and competed with a .226, 9mm, and 12 ga?
That would be perfect for factory or practical division.
How many magazines do you need to carry with a .45 if you're only allowed a max of 10 per?
Depends on the stage design but you can carry as many as necessary to complete the stage. I usually carry 2 extra on my hip and one in a pocket in case I really suck. But if the stage is handgun-heavy I'll carry more.
What is the typical magazine for a rifle (that's less than .308)?
Factory you have to run standard 30-round mags without any magazine basepads that add length. Practical you see a lot of 30-rounders but some run magpul 40 rounders and I saw one guy with an X-15 50-round drum mag at the match. IME it's rare to see a stage that requires more than 30 rounds of rifle shots at a time.
Other things you might be interested in: the standard 3GN square paper target has a square outer perforation and a round inner perforation. To neutralize that target you have to EITHER hit it twice anywhere inside the outer perf or once in the center perf. Most everyone double taps them every time just to make sure, because it's quicker to throw another shot at it than to take a miss if you don't hit it once in the center. (However, HEAVY division gets a scoring concession where you only have to hit it once anywhere inside the perf to neutralize it with any gun... this helps me because .45 and .308 or .30-06 -- which I run in my Garands -- is much more expensive than .223 and 9mm).
Steel knock-over targets are generally optional shotgun or pistol, unless they're past 40 or 50 yards away in which case they're almost always rifle only. Generally they don't place steel rifle targets closer than about 80 yards or so.
Some clubs require the use of buckshot on some targets; mine doesn't. Birdshot or slugs only, and they generally have only one or 2 stages requiring slugs, and usually only one or 2 slugs per stage (they're expensive). So ask around your area where you plan to shoot. People are generally very nice, friendly, and helpful. More than once I've seen people loaning firearms to a competitor who had a breakage or whatever, and I've given a lot of ammo away to new shooters who didn't realize the "suggested round count" doesn't include all their misses.

You can't run any ammo that is magnetic (the actual bullet can't attract a magnet) or use steel shot in your shotgun.