We study the systematics of a d-dimensional lattice model for melts of semiflexible living polymers. For d=2 and 3 our model, which includes vacancies, loops, and the possibilities of polymerization and polydispersity, exhibits both equilibrium crystallization and glass formation in the wake of a quench. We study these analytically, in some limits, and via extensive Monte Carlo simulations. A continuous Ising-type transition separates crystalline and disordered phases for the d=2 square lattice. If loop formation is favored in d=2, crossover effects lead to power-law decays of polymer-length distributions over large length scales, strong fluctuations in thermodynamic quantities, and slow relaxation. These crossover effects arise because of the proximity of a phase with an infinite correlation length in one limit of our model. For the d=3 simple cubic lattice our model has a first-order crystallization transition. Quenches from the disordered to the ordered phase yield glassy, metastable configurations for both d=2 and 3. We study the latter case in: detail and find logarithmically slow relaxation out of these metastable configurations, a frustration-driven glass-crystal transition, and an exotic lamellar glass. We propose a Monte Carlo:analog of scanning calorimetry and use it to study these glasses. We discuss the relevance of our work to experiments on different systems of living polymers, earlier studies of crystallization in polymeric melts, and some theories of the glass transition in model systems. [S1063-651X(99)07201-3].