Is there a set of NP-hard problems that are most commonly used for proving NP-hardness?
Most textbooks will have a set of 'core' NP-Complete problems from which all their examples and exercises would reduce from, which are probably the ones you list (3SAT, Vertex Cover, Independent Set, Clique, k-Coloring, Set Cover, Dominating Set, Hitting Set, Subset Sum, Knapsack.)

A bigger set of 'standard' problems would be Garey &amp; Johnson's book. Beyond that, people look at hard restrictions of the standard problems (i.e. k-coloring for k=3, Exact Cover version of set cover, not-all-equal 3SAT, etc).

Beyond that, it gets fairly domain specific.