Magnetoelasticity plays a crucial role in numerous magnetic phenomena, including magnetocalorics, magnon excitation via acoustic waves, and ultrafast demagnetization, or the Einstein-de Haas effect. Despite a long-standing discussion on anisotropy-mediated magnetoelastic interactions of relativistic origin, the exchange-mediated magnetoelastic parameters within an atomistic framework have only recently begun to be investigated. As a result, many of their behaviors and values for real materials remain poorly understood. Therefore, by using a proposed simple modification of the embedded cluster approach that reduces the computational complexity, we critically analyze the properties of exchange-mediated spin-lattice coupling parameters for elemental 3d ferromagnets (bcc Fe, fcc Ni, and fcc Co), comparing methods used for their extraction and relating their realistic values to symmetry considerations and orbitally decomposed contributions. Additionally, we investigate the effects of noncollinearity (spin temperature) and applied pressure on these parameters. For Fe, we find that single-site rotations, associated with spin temperatures around 100 K, induce significant modifications, particularly in Dzyaloshinskii-Moriya-type couplings; in contrast, such interactions in Co and Ni remain almost configuration independent. Moreover, we demonstrate a notable change in the exchange-mediated magnetoelastic constants for Fe under isotropic contraction. Finally, the conversion between atomistic, quantum-mechanically derived parameters and the phenomenological magnetoelastic theory is discussed, which can be a useful tool towards larger and more realistic dynamics simulations involving coupled subsystems.