It's possible that Mr. Squiggles is older than he seems but it seems more likely that bad genetics and bad petstore care are the real culprits. Aside from getting sick, he's showed no other typical signs of aging like color changing or cataracts. Those signs usually don't show up until a betta is over a year old, usually closer to two or three.
There is a possibility the other two could get sick, yes. Ideally, we should all have separate buckets and siphon hoses for each tank we have so we never cross-contaminate. In the real world, that's usually not even remotely possible for most of us (myself included). The only time this kind of practice becomes an absolute must is if a person is a breeder and has a fish room full of extremely valuable fish. For the rest of us, this is the risk we run. 9 times out of 10, nothing happens but on some rare occasion, if a fish gets stressed and his immune system crashes, he could get sick from pathogens introduced via the siphon or shared decor.
You may want to sterilize the siphon hose using hot water. Otherwise, another preventative measure you could take is to put both fish in either a 24 hour soak of methylene blue or 1 tsp of aquarium salt per gallon for a few days. Methylene blue is a good disease preventative for both internal and external problems because it absorbs well into the system but it stains everything and can be hard to find. If you've ever seen the bettas in the blue water cups, they're in methylene blue. The AQ salt would work in a similar way but only externally. It would encourage them to produce slime coat and help regulate their osmotic functions. We just can't have Lemongrab and the rescue get sick. We just can't! *puts foot down*