Dynamic Monte Carlo simulations of single star-branched polymer models were made. Star macromolecules were confined to a simple cubic lattice with the nearest-neighbor attractive interactions. Every star consisted of f=3 arms of equal length. Length of a star varied between 49 and 799 statistical segments. Static and dynamic properties of model stars were calculated in good solvent conditions, Theta-state and in the collapsed state. Change of the chain dimensions, diffusion coefficients, and their scaling exponents with the temperature was shown and discussed. The locations of the Theta temperature and the collapse transition temperature T-C were estimated for all chain lengths under consideration. The differences in motion of inner and outer parts of a star-branched polymer in different solvent conditions were described. The existence of a high-density core in the center of star macromolecules was confirmed and the influence of the temperature on its magnitude was studied. An analysis of motion of different parts of star polymers and of the number of inter- and intra-armal contracts was performed.