Performing dynamic off-lattice multicanonical Monte Carlo simulations, we study the statics, dynamics, and scission-recombination kinetics of a self-assembled in situ-polymerized polydisperse living polymer brush (LPB), designed by surface-initiated living polymerization. The living brush is initially grown from a two-dimensional substrate by end-monomer polymerization-depolymerization reactions through seeding of initiator arrays on the grafting plane which come in contact with a solution of nonbonded monomers under good solvent conditions. The polydispersity is shown to significantly deviate from the Flory-Schulz type for low temperatures because of pronounced diffusion limitation effects on the rate of the equilibration reaction. The self-avoiding chains take up fairly compact structures of typical size R-g(N) similar to N-nu in rigorously two-dimensional (d = 2) melt, with nu being the inverse fractal dimension (nu = 1/d). The Kratky description of the intramolecular structure factor F(q), in keeping with the concept of generalized Porod scattering from compact particles with fractal contour, discloses a robust nonmonotonic fashion with q(d)F(q) similar to (qR(g))(-3/4) in the intermediate-q regime. It is found that the kinetics of LPB growth, given by the variation of the mean chain length, follows a power law < N(t)> proportional to t(1/3) with elapsed time after the onset of polymerization, whereby the instantaneous molecular weight distribution (MVVD) of the chains c(N) retains its functional form. The variation of < N(t)> during quenches of the LPB to different temperatures T can be described by a single master curve in units of dimensionless time t/tau(infinity), where tau(infinity) is the typical (final temperature T-infinity-dependent) relaxation time which is found to scale as tau(infinity) proportional to < N(t = infinity)>(s) with the ultimate average length of the chains. The equilibrium monomer density profile phi(z) of the LPB varies as phi(z) proportional to phi(-alpha) with the concentration of segments phi in the system and the probability distribution c(N) of chain lengths N in the brush layer scales as c(N) proportional to N-tau. The computed exponents alpha approximate to 0.64 and tau approximate to 1.70 are in good agreement with those predicted within the context of the Diffusion-Limited Aggregation theory, alpha = 2/3 and tau = 7/4.