Re: Can you be exposed to doing something?
I agree that the sentence, without context, could have been written better and I would take it to mean:
"Children are exposed to lifestyles in which relationships break (breaking ties) and other relationships that are probably destined to fail (creating insecure attachment styles.)"
However, it is possible that "breaking ties and creating insecure attachment styles" have been referenced earlier in the piece, in which case, the sentence may be OK.
"There are no rules in English, only guidance. Some guidance looks like a rule; it probably isn't."