We present a configurational-biased lattice Monte Carlo scheme for simulating nonideal dendrimers that satisfies detailed balance. This corrects an important shortcoming in a previous lattice Monte Carlo scheme by Mansfield and Klushin: in a previous publication, we showed that the Mansfield and Mushin scheme did not obey detailed balance, and that this led to surprisingly large errors in the radius of gyration R-g and scattering form factor P(q) for ideal dendrimers. In this paper, we have calculated the radius of gyration, the form factor, and the intramolecular density profile for g = 1-8 self-avoiding dendrimers and find that our results are qualitatively the same as previous results obtained by Mansfield and Klushin (g is the generation number). This indicates that the error in the Mansfield and Klushin scheme due to detailed balance violation is much smaller for self-avoiding dendrimers. Our other key conclusions concerning the equilibrium properties of self-avoiding dendrimers are the following: (1) The radius of gyration scales with the total number of monomers roughly as R-g proportional to N-1/3. A more careful analysis however shows that there is a small (similar to30%) and nonmonotonic variation in the internal density of the dendrimer with N, the form and magnitude of which are consistent with previous intrinsic viscosity results on dendrimers. (2) The intramolecular density profile is dense core like at low dendrimer generations (g < 5) and solid sphere like at high generations (g greater than or equal to 5). (3) There is some "hollowness" in the core region of the dendrimer for higher generation dendrimers (g greater than or equal to 5), though the depth and extent of the hollowness are much smaller than that predicted by the dense shell model. (4) Terminal groups of the dendrimer are not localized at the periphery but delocalized throughout the dendrimer. The relationship between our findings and previous theoretical and experimental studies, especially recent scattering studies on dendrimers, is discussed.