Does this mean we're not responsible for something happening, or we aren't responsible ie. trustworthy in general?
Depends on the context, if you are talking about something in particular, or just how a person is. Ex.: eu não sou responsável pelo cavalo / eu não sou uma pessoa responsável.
Both, but I think people would say it like "Você é irresponsável" to mean not being trustworthy. Most situations where the word responsável appears, it will be referring to something in particular.
Yes, it may have both meanings.
Could this also mean "We are not to blame? "
I wonder if this is commonly used by teens in Brazil too.