Commonly mistaken as an Enduring Power of Attorney, a Power of Attorney is a document that gives another person, an Attorney, the authority to act on your behalf in certain circumstances. Generally, a Power of Attorney will only apply while someone is out of the country or physically incapacitated. Enduring Powers of Attorney give an Attorney the authority to manage your property, and personal care and welfare matters, but is not revoked if you lose your mental capacity.
The current Trustee Act 1956 is silent as to whether beneficiaries are entitled to disclosure of particular trust information and documents. When the law is not clear, the Courts determine what would have been intended by Parliament when passing that particular law. This is exactly what has happened in the recent Supreme Court case Erceg v Erceg  NZSC 28.
The Care of Children Act 2004 makes provision for a parent to appoint Testamentary Guardians for their children through their will. If both parents (or one parent in some situations) die without a Testamentary Guardian appointed, it will be a decision of the Courts to appoint a guardian for your children. There is no guarantee that your child/children will be appointed a Testamentary Guardian that you would have chosen yourself.
A beneficiary of a Trust does not have a right (as such) to information held by the Trust. Beneficiaries can request information from trustees, however, trustees can refuse to provide it in exercising their fiduciary duties. But what does that mean/what process do they have to go through in deciding whether or not to disclose information? If the new Trusts Bill is enacted, the law around disclosure of information to beneficiaries will be made clearer.