Temporal-dissipative Kerr solitons are self-localized light pulses sustained in driven nonlinear optical resonators. Their realization in microresonators has enabled compact sources of coherent optical frequency combs as well as the study of dissipative solitons. A key parameter of their dynamics is the effective detuning of the pump laser to the thermally and Kerr-shifted cavity resonance. Together with the free spectral range and dispersion, it governs the soliton-pulse duration, as predicted by an approximate analytical solution of the Lugiato-Lefever equation. Yet a precise experimental verification of this relation has been lacking so far. Here, by measuring and controlling the effective detuning, we establish a way of stabilizing solitons in microresonators and demonstrate that the measured relation linking soliton width and detuning deviates by less than 1% from the approximate expression, validating its excellent predictive power. Furthermore, a detuning-dependent enhancement of specific comb lines is revealed due to linear couplings between mode families. They cause deviations from the predicted comb power evolution and induce a detuning-dependent soliton recoil that modifies the pulse repetition rate, explaining its unexpected dependence on laser detuning. Finally, we observe that detuning-dependent mode crossings can destabilize the soliton, leading to an unpredicted soliton breathing regime (oscillations of the pulse) that occurs in a normally stable regime. Our results test the approximate analytical solutions with an unprecedented degree of accuracy and provide insights into dissipative-soliton dynamics.