I went to the My Family page and it was empty.  I have two children in the group.  Why don't I see them there?

When your group entered your children's membership, they did not create the link between your children's membership records and yours.

A leader who is authorized for the Membership role can correct this for you by going to Membership ? Active Members and selecting your record.  At the bottom of the page is a grid showing the youth members to whom you are related and an Add A New Item button to add the ones that are missing.

I entered a new password and pressed the Save button and nothing seemed to happen.  Why is that?

The screen was redisplayed after it saved your new password.  That is how this function works.

Can a youth member sign up for an event?

Yes.  Both youth and adults have access to the Sign Up For Events page.