We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space Q_g,nof genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers ∫-_Mg′,n′Ψ^d1₁···Ψ^dn′_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 N_g′,n′(b₁,⋯,b_n′) that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces M_g′,n′(b₁,⋯, b_n′) of bordered hyperbolic surfaces with geodesic boundaries of lengths b₁,⋯,b_n′. 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 Mod_g,n·γ of any simple closed multicurve γ inside the ambient set ℳℲ_g,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 Q_g,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.