The coupling between antiferromagnetic spins and infrared-active phonons in solids is responsible for many intriguing phenomena and is a field of intense research with extensive potential applications in the modern devices based on antiferromagnetic spintronics and phononics. Insulating rutile antiferromagnetic crystal ${\mathrm{CoF}}_{2}$ is one of the model materials for studying nonlinear magnetophononics due to the strong spin-lattice coupling as a result of the orbitally degenerate ground state of ${\mathrm{Co}}^{2+}$ ions manifested in the plethora of static and induced piezomagnetic effects. Here we report results on the complete infrared spectroscopy study of lattice and magnetic dynamics in ${\mathrm{CoF}}_{2}$ in a wide temperature range and their careful analysis. We observed that infrared-active phonons demonstrate frequency shifts at the antiferromagnetic ordering. Furthermore, using first-principles calculations, we examined the lattice dynamics and disclosed that these frequency shifts are rather due to the spin-phonon coupling than geometrical lattice effects. Next we found that the low-frequency dielectric permittivity demonstrates distinct changes at the antiferromagnetic ordering due to the spontaneous magnetodielectric effect caused by the behavior of infrared-active phonons. In addition, we have observed magnetic excitations in the infrared spectra and identified their magnetodipole origin. To strengthen our conclusions, we analyze the theoretical phonon-magnon coupling overall phonons at the $\mathrm{\ensuremath{\Gamma}}$ point. We conclude that the largest effect comes from the ${A}_{1g}$ and ${B}_{2g}$ Raman-active modes. As such, our results establish a solid basis for further investigations and more deeper understanding of the coupling of phonons with spins and magnetic excitations in antiferromagnets.