We study the formation, growth, and co-evolution of single and multiple supermassive black holes (SMBHs) and compact objects like neutron stars, white dwarfs, and stellar mass black holes in galactic nuclei and star clusters, focusing on the role of stellar dynamics. In this paper we focus on one exemplary topic out of a wider range of work done, the study of orbital parameters of binary black holes in galactic nuclei (binding energy, eccentricity, relativistic coalescence) as a function of initial parameters. In some cases the classical evolution of black hole binaries in dense stellar systems drives them to surprisingly high eccentricities, which is very exciting for the emission of gravitational waves and relativistic orbit shrinkage. Such results are interesting to the emerging field of gravitational wave astronomy, in relation to a number of ground and space based instruments designed to measure gravitational waves from astrophysical sources (VIRGO, Geo600, LIGO, LISA). Our models self-consistently cover the entire range from Newtonian dynamics to the relativistic coalescence of SMBH binaries.