They may be law, but they indeed are NOT actually part of the Constitution -- in this case it is UN law, which has no authority to enforce this treaty.
No. A binding treaty will override the Constitution if that is the necessary effect of the treaty's Language. If that effect is no longer desired, then the US can either renegotiate the treaty or it can withdraw from the treaty, but until then, the treaty would control.
Thus, if the US signs, and ratifies, a treaty in which the US promises that it will not do something that it is otherwise entitled to do, that promise is binding. Since a treaty is frequently made to benefit certain third parties, any one of those people would notmally have standing to enforce the treaty.
A simple example would be the income tax treaties the US has with many other countries. Under the 16th Amendment, Congress has the power to tax income from whatever source derived. However, in its tax treaties, the US has given up that power in certain ways. For example, if a treaty resident operates a business in the US, the US cannot tax the income from that business unless the income is attributable to a permanent establishment.
If the US nonetheless tries to tax income over which it has given up its taxing power, the individual being taxed can sue the US government to have the assessment abated and any collected tax refunded.
One thing to look out for in many treaties is whether enabling legislation is required. Some treaties, like the tax treaties, are self-executing, in the sense that the treaty is operative once ratified, without the need for legislation to put it into effect. Other treaties require enabling legislation, and until that is passed the treaty generally has no real effect.
