Are parents responsible for their children's actions and to what extent should they be responsible? (See details)

Inspired by a discussion that started in this question. I’m not asking about legal responsibility, just what each of us views as the personal responsibility of parents.

If a child breaks something at someone else’s house or at a store, is the parent responsible for replacing that item? Does it make a difference if it was broken on purpose or on accident?

If a child knocks someone else down, is the parent responsible for making sure the person is okay? Should the parent cover any medical costs the person encounters (whether it be due to lack of insurance or because they have crappy insurance and they are responsible for a lot of the cost out of pocket)? Does it make a difference if this happened on accident or on purpose?

Are parents responsible for teaching their children that what they did can have an effect on others (such as breaking things or hurting other people)? Does this responsibility change if the thing the child did was an accident or on purpose?

Are there particular ages you feel a parent is responsible for their child’s actions? Are there particular ages when they are not responsible for their child’s actions?