Any somewhat active mixed breed will do. There are poodle mixes, Shi Tzu mixes, lab mixes, shepherd mixes, pit mixes, and Heinz 57s at our club. The vast majority seem to love it. My own dog is reportedly an Australian Shepherd/Collie/maybe lab mix (I call him a herding mutt) and he's good at it, he's fast, and he loves it. So, really, it depends on how serious you want to get and how good you want to be.
Your herding dogs are going to be your go-to breeds to look for. Border collies and mixes thereof are generally the top dogs at agility competitions. They are blazing fast and respond to a literal drop or raising of your shoulder. If I was going to gauge wether they have any drive for it or not, I would be rolling a ball on the floor for them and see how interested they are in chasing it. Their herding instinct often translates into a really excellent agility ability.
HOWEVER. Getting a herding breed from a rescue can be a gamble. SO MANY have "stranger danger" issues, reactivity issues, are afraid of men, or anxiety issues. So, they may be great and love to do agility at home or in a small group, but when they get into the hubbub of a trial, they shut down or perform like crap. My own dog is reactive and has stranger danger issues, but I lucked out incredibly because for some reason that all goes away when we do agility. He suddenly becomes a dog that gives not a fart about anything around him, handles dogs screaming at him and a loose dog ramming her nose up his butt, and actually shows interest in meeting new people because they might have a treat for him. BUT, because of his issues outside of the agility environment, it makes traveling or staying outside of our home for agility trials difficult.
There are also many herding dogs who are OUTSTANDING in practice where there are few people and little noise, but when they enter the trial environment they are far too overwhelmed and just completely shut down, or they become incredibly reactive and even dangerous.
So, in a nut shell, if you want to go with a herding breed from a rescue, I would choose either an adult with a set personality who has been in foster long enough for the foster to gauge their personality, or I would go with a reputable breeder who has consistently produced decent performance dogs and can assure me their dogs won't develop "stranger danger" and severe anxiety. There are even a number of breed specific rescues who have adult dogs that may have taken an agility class or two, or even already have titles.
I've also seen a number of really incredible lab/lab mixes who are both quick and accurate, but they are on the leaner, longer legged side of the lab spectrum. Small shepherd mixes are also some of the stand outs at trials. They are usually not particularly fast, but they clearly enjoy the game. Small terrier mixes are speedy little things who love the game, too, but their owners report they can be independent and sometimes just want to do their own thing.
If I were going into a shelter looking for an agility dog, I would avoid the Shi Tzu mixes or the small, independent companion type dogs. I've seen a number do agility, but I feel like their owners have to work really hard to keep them interested, and sometimes no matter how much encouragement they provide they end up wandering off. They generally seem to enjoy the game and are so fun to meet and play with outside the ring, but when they don't want to play the game...they don't!
As for games for your potential senior dog, depending on health, they do have "Preferred" in agility which allows the dog to jump 4 inches shorter than their required height based on their height measurement. So if your dog measures 23 3/4 inches tall and is supposed to jump 24 inches, they can jump 20 inches in preferred classes.
Rally or scent-work would also be options.