If you don't have 50-50 custody now, you'd need to show a court why things have changed, since the last court order was issued, in a way that makes this the right thing to do now, and that the change of custody is in the children's best interests.  This probably isn't the sort of case that you can present effectively, without a lawyer, because the decisions are usually very sensitive to the detailed facts of the particular people involved.

Responsibility for healthcare usually depends on the money and insurance available to the parents.