No, not at all. 'Do-gooder' is a fairly negative term. If you call someone a 'do-gooder', this means that you don't think that the help they are giving is useful. You might even be irritated by the fact that they are interfering. 'Volunteer' is a neutral term. It simply means that someone offers to do a task without payment.

No, they are not. A volunteer is somebody who does some work without being required, normally on an unpaid basis. Do-gooder has definite negative connotations, it is somebody who believes that, or acts as if they are, morally superior to others because of their voluntary work, or goes about it in a bit of a condescending way. It is normally a bit of an insult to call someone a do-gooder.