Second question first:mostly22lately, thanks. For Rimfire Challenge, what pistols and rifles do you and your wife like and dislike?
For the Rimfire Challenge Limited Class, can you use aftermarket triggers, like the TG2000 or Kidd?
In both classes, there aren't any operational restrictions about triggers and whatnot. The firearms that are most competitive are often 'idealized' for speed and if that means a specialty trigger, so be it. Rimfire Challenge also has a general "run what you brung" attitude because it is, foremost, a newcomer-welcoming environment.
It's better to arrive at events and participate instead of focusing on equipment. Because it is also family-oriented, it is completely common to see two and even three generations of one family participating, and sometimes using the same two guns! Somebody on the back table has extra magazines and is reloading while the other one is at the line.
Now, the first question...
I hesitate to list gear for others to consider. Tons of videos out there showing rimfire challenge events to watch and see for yourself. What works for us (and that changes as we learn and adapt) won't be the best choice for others.
Because we are actively competing open, and squad with people who have sponsorships and regularly attend national competitive matches, we are using idealized equipment to try to help us along as we're just a bit older than some of the fast folks. I share this context because to answer the question directly could scare folks off due to the expense. The vast majority of people are there to have fun and having fun emphatically does NOT require dropping massive dollars. That kind of big $$ thing is for other shooting sports.
In this sport, reliability / function is #1, then optics. Neither of these require major expenditures, in relative terms. Your goal is to go fast and not miss. Can you hit a 8" or 10" plate at 10 yards? Remember, groupings here are meaningless. All you gotta do is hit the thing and move to the next one. Simple, right?
You shooting 7 or 8 rounds in 3 seconds, five times in a row, means your gear has to do that too. Ammo becomes a key part of the equation, at least as important as the gun. You don't run Thunderbolts; neither do you bring Lapua. Unless those are all you have!
The other participants we compete with have a wide variety of gear, and numerically I'd say ... the pistols are more than half various flavors of Volquartsen parts or assemblies. The majority of the rifles are some versions of Ruger 10/22, with varying upgrades for reliability, weight and balance not uncommon.
For pistol and rifle in open class, the most successful people have paid attention to the optics that work best for their eyes. For limited, the sights selection is even MUCH more personally dependent on one's own vision.
I don't "dislike" any firearm. I will be honest though, I do share the disappointment others have when they have equipment malfunctions. It happens to me, too.
So I learned what causes this (often the hard way) and try my best to make sure it does not.
Also, I know I can't move as quickly with some of my gear as I can with others. I don't use my CZ457 at Rimfire Challenge, nor my S&W 41, even though in any other context these are truly fine 22lr firearms. I also don't use anything with a scope, or a tubular magazine. Who needs the handicap? You're probably getting the idea by now.
Come to an event or two, and shadow a squad as they go between stages. That's the best advice I can offer.