1. What is the Constitutional basis for gun ownership?
The Constitutional issues of ownership are contained explicitely in these sections.
The basic Constitution gives Congress the power to regulate the Militia.
The Constitution of the United States of America
Article I.
Sect. 8. The Congress shall have power

To raise and support armies, but no appropriation of money to that use shall be for a longer term than two years;

To provide and maintain a navy;

To make rules for the government and regulation of the land and naval forces;

To provide for calling forth the militia to execute the laws of the union, suppress insurrections and repel invasions.;

To provide for organizing, arming, and disciplining, the militia, and for governing such part of them as may be employed in the service of the United States, reserving to the States respectively, the appointment of the officers, and the authority of training the militia according to the discipline prescribed by Congress;

The 2nd amendment limits the basic power of Congress in regards to the militia in that it prohibits the power of Congress in regard to the militia from disarming the militia. (Noting at the time the militia was considered to be every able bodied male, etc.)
2nd Amendment
A well-regulated militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed.
The 14th prohibits the denial of an Federal right by any state. (The Doctrine of Incorporation holds the 2nd has not been incorporated as there has not been any Supreme Court decision related to a specific state law.)
14th Amendment
Sect. 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law, nor deny any person within its jurisdiction the equal protection of the laws.