We investigate key aspects of coherent optical communications on inter-satellite links (ISLs) for the next-generation ultra-dense low-Earth orbit (LEO) constellations. Initially, the suitability of QPSK, 8-QAM, and 16-QAM modulation formats with different symbol rates (28 GBaud, 60 GBaud, and 120 GBaud) and channel coding schemes (oFEC and staircase codes) for intra- and interorbital connections is evaluated. We provide SNR margins for all investigated sets and determine unfeasible operating points. We show that sets with higher-order modulation formats combined with high symbol rates can prove unfeasible, even for first-neighbor connections. Furthermore, the presence or absence of optical pre-amplification as well as the choice for a more robust channel coding technique, such as the oFEC, can be decisive in certain LEO-LEO links. Next, we characterize the Doppler shift (DS) and its time derivative for first-neighbor interorbital connections in two different topologies and for general connections established between any pairs of satellites. Our results reveal that while the maximum Doppler-generated frequency shift amplitude can be considerably higher than those typically found in fiber-optic communications, the time derivative values are significantly lower. Finally, we address all-digital DS compensation in extreme cases of frequency offset amplitude and derivative where the typical Mth-power algorithm is not sufficient. To this end, we propose a filtered version of an existing two-stage method combining spectral shifts with the Mth-power method. The simulation results indicate that this approach provides an appropriate solution for all examined cases.