We present results from a study of X(3872) -> pi pi J/psi decays produced via exclusive B -> KX(3872) decays. We determine the mass to be M-X(3872) = (3871.85 +/- 0.27(stat) +/- 0.19(syst)) MeV, a 90% confidence level upper limit on the natural width of Gamma(X(3872)) < 1.2 MeV, the product branching fraction B(B+ -> K+X(3872)) X B(X(3872) -> pi(+)pi(-) J/psi) = (8.63 +/- 0.82(stat) +/- 0.52(syst)) X 10(-6), and a ratio of branching fractions B(B-0 -> (KX)-X-0(3872))/B(B+ -> K+X(3872)) = 0.50 +/- 0.14(stat)) +/- 0.04(syst). The difference in mass between the X(3872) -> pi(+)pi(-) J/psi signals in B+ and B-0 decays is Delta M-X(3872) = (-0.71 +/- 0.96(stat) +/- 0.19(syst)) MeV. A search for a charged partner of the X(3872) in the decays <(B)over bar>(0) -> K-X+ or B+ -> (KX+)-X-0, X+ -> pi(+)pi(0) J/psi resulted in upper limits on the product branching fractions for these processes that are well below expectations for the case that the X(3872) is the neutral member of an isospin triplet. In addition, we examine possible J(PC) quantum number assignments for the X(3872) based on comparisons of angular correlations between final state particles in X(3872) -> pi(+)pi(-) J/psi decays with simulated data for J(PC) values of 1(++) and 2(-+). We examine the influence of rho-omega interference in the M(pi(+)pi(-)) spectrum. The analysis is based on a 711 fb(-1) data sample that contains 772 x 10(6) B (B) over bar meson pairs collected at the gamma(4S) resonance in the Belle detector at the KEKB e(+)e(-) collider.