All those questions can be answered easier if you have some premise for defining what a natural-right or government-right is. Do you?
Here are some definitions. Which do use?
any right that exists by virtue of natural law.http://dictionary.reference.com/browse/natural+right
Natural and legal rights are two types of rights theoretically distinct according to philosophers and political scientists. Natural rights are rights not contingent upon the laws, customs, or beliefs of any particular culture or government, and therefore universal and inalienable. In contrast, legal rights are those bestowed on to a person by the law of a particular political and legal system, and therefore relative to specific cultures and governments.http://en.wikipedia.org/wiki/Natural_and_legal_rights
1. General: Fundamental human rights based on universal natural law, as opposed to those based on man-made positive law. Although there is no unanimity as to which right is natural and which is not, the widely held view is that nature endows every human (without any distinction of time or space, and without any regard to age, gender, nationality, or race) with certain inalienable rights (such as the right to 'life, liberty, and pursuit of happiness') which cannot be abrogated or interfered with by any government. And that, whether or not these rights are enshrined in a national legal code, no government is lawful if it fails to upholds them. See also human rights.http://www.businessdictionary.com/definition/natural-rights.html
rights which persons possess by nature: that is, without the intervention of agreement, or in the absence of political and legal institutions. Natural rights are therefore attributable to individuals without distinction of time or place. A contrast may be drawn with positive rights: that is, those rights conferred or guaranteed by a particular legal system. Natural rights have been derided as nonsensical (by Bentham) on the ground that it is impossible to speak of rights without enforceable duties, and enforceability exists only when a potentially coercive legal system exists. Furthermore, there has been no unanimity even amongst those who recognize natural rights as to their content. Natural rights have been seen as gifts of God, as correlative to duties imposed on man by God, and as concomitants of human nature or reason. We might distinguish: (1) natural rights; (2) moral rights; and (3) legal rights. The third are those recognized by positive law. The first are those asserted to be universal and thus guides to the proper content of any legal system. The second are those which, it is claimed, should be recognized by particular legal systems or which, while not universal, should be recognized under existing conditions. The classification of rights will depend in part on understandings of their purpose and of their consequences.http://www.answers.com/topic/natural-right