We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space Qg,nof genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers ∫-Mg′,n′Ψd11···Ψdn′n′with explicit rational coefficients, where g′ <g and n′ <2g+n. The formulas obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomials Ng′,n′(b1,⋯,bn′) that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces Mg′,n′(b1,⋯, bn′) of bordered hyperbolic surfaces with geodesic boundaries of lengths b1,⋯,bn′. A similar formula for the Masur-Veech volume (but without explicit evaluation) was obtained earlier by Mirzakhani through a completely different approach. We prove a further result: The density of the mapping class group orbit Modg,n·γ of any simple closed multicurve γ inside the ambient set ℳℲg,n(ℤ) of integral measured laminations, computed by Mirzakhani, coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to γ among all square-tiled surfaces in Qg,n. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when n = 0. In particular, we compute explicitly the asymptotic frequencies of separating and nonseparating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g, and we show that in large genera the separating closed geodesics are √2/3πg · 1/4g times less frequent.