Based on the general polytropic hydrodynamic formalism of self-similar gravitational collapse with spherical symmetry, we study the self-similar dynamic formation of black holes (BHs) in several astrophysical and cosmological contexts such as stellar mass BHs (sMBHs), intermediate mass BHs (IMBHs), supermassive BHs (SMBHs) and hypermassive BHs (HMBHs) -the latter two in the early Universe are of special interest here. We invoke the Paczynski-Wiita gravity to capture the key effects of general relativity and show the growth of BHs including SMBHs and HMBHs can be very rapid and effective in sufficiently massive host halos within timescales shorter than the Universe age of ∼ 13.8 billion years. The effective pressure in our hydrodynamic formalism can represent that of gas in stars, that of random stellar motions in globular clusters, in bulges of spiral galaxies, and in elliptical galaxies, that of random galaxy motions in clusters of galaxies and that of random motions of dark matter (DM) particles as well as that of combinations of various pertinent components (including turbulence). In particular, we have predicted in 2013 that HMBHs in the mass range of ∼ 10 10 − 10 12 M ⊙ or larger can exist in the Universe and even in the early Universe in the forms of hard X-ray/gamma-ray sources or quasars [1]. We discuss several observational evidence for pertinent aspects and classifications and further speculate the possibility of DM BHs (DMBHs) and mixed matter BHs (MMBHs) in reference to the Bonnor-Ebert critical equilibrium condition with two tentative observed candidates. Based on the three-dimensional perturbation analysis of a spherical dynamic collapse [2,3], certain internal gravity modes (g-modes) and all vorticity modes (v-modes) are unstable, implying inevitable turbulent massive motions during a dynamic collapse of forming a BH. Physical consequences including gravitational wave emissions, magnetohydrodynamic (MHD) dynamo processes, and acceleration of ultra-high energy cosmic rays (UHECRs) associated with such massive turbulence during dynamic collapses are also discussed. For the Southern African Large Telescope (SALT) further strengthened with infrared spectrometers, it is certainly capable of identifying quasars and ultra and extremely luminous infrared galaxies harboring SMBHs and/or HMBHs in the Universe including the early Universe. It would be of considerable interest to study the neutral hydrogen environments of SMBHs and HMBHs with increasing redshift z in a systematic manner.