You would more often hear "the responsible thing to do" and it is not the same as "responsible for" If somebody is responsible for something, it means that either: 1) That person has been assigned that task or is answerable for the successful conclusion of the task or the well-being of the thing. 2) The person is assigned credit (or blame) for something that happened. "the responsible thing to do" does not assign responsibility to a given person or persons. It simply states that that 'thing' (whatever it is) would demonstrate maturity and a sense of responsibility for the person doing it. Cleaning up after the party is the responsible thing to do.

No!

I think "In charge of something" is more frequently used if you mean 承担