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We show that the existence of constant scalar curvature K\"ahler (cscK)metrics with cone singularities is equivalent to the properness of log$K$-energy. We also prove their equivalence to the geodesic stability. They areextensions of the solution of the properness conjecture and Donaldson'sgeodesic stability conjecture of the cscK problem for smooth K\"ahler metricsby Chen-Cheng to the setting of cscK cone metrics.  One applications of our main results is that we introduce and constructsingular cscK metrics with possible degeneration in big cohomology class. Asanother application, we also prove both openness and approximation property forthe path of cscK cone metrics, which are paralleling to Donaldson's continuitymethod through K\"ahler-Einstein cone metrics in the resolution ofYau-Tian-Donaldson conjecture for Fano K\"ahler-Einstein metrics.

Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stability