My condolences on your grandma. I hope you're ok.
I thought you only adopted 2, why are there 3 now?
Anyhow, I can't say for certain, but I don't think it was the puff of air that caused them to be frightened. The puff of air doesn't actually do any damage, and it's not scary or anything, just a minor irritation. (which is why people recommend it)
My theory: you left your hand in the cage too long, and you went over their comfort zone. They felt that you had overstayed your welcome, and crossed into "intruder" zone, and the nibbles were to warn you away, and when the daughters got frightened the mum came out, and when you still didn't get the message, she bit you SERIOUSLY.
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Chins (and dogs cats etc) do sometimes bite or scratch too hard, BUT, generally, all animals know that when they draw blood, they have crossed a certain line. Any time an animal bites hard enough to cause bleeding (and you said it was bleeding "a lot") they know that they have caused injury. If an animal tastes blood in it's mouth and still doesn't let go, it is NOT PLAYING AROUND, it knows that it is causing harm and is doing it on purpose.
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Having crossed their comfort zone and intruded on their personal space, you should be careful not to do it again, ESPECIALLY not too soon... as grayrodent said, give them time.
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Personally though, what I would try? ... my theory is that chins are surprisingly intelligent. They sometimes understand underlying subtleties that would surprise you.
Try a peace offering/ apology.
Every day, go to their cage, gently and slowly apologize for disturbing them, and place their treats in their cage, then back off to the other end of your room. Wait for them to take the treat. After they take it, gently say "Thank you... I hope we can be friends again." Then exit the room.
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It might feel silly doing this for a little animal that doesn't speak english, but try it out and see if it works. You might be very very surprised!