Normalised multi-differential cross sections for top quark pair (tt^-) production are measured in proton-proton collisions at a centre-of-mass energy of 13 TeV using events containing two oppositely charged leptons. The analysed data were recorded with the CMS detector in 2016 and correspond to an integrated luminosity of 35.9fb^-1. The double-differential tt^- cross section is measured as a function of the kinematic properties of the top quark and of the tt^- system at parton level in the full phase space. A triple-differential measurement is performed as a function of the invariant mass and rapidity of the tt^- system and the multiplicity of additional jets at particle level. The data are compared to predictions of Monte Carlo event generators that complement next-to-leading-order (NLO) quantum chromodynamics (QCD) calculations with parton showers. Together with a fixed-order NLO QCD calculation, the triple-differential measurement is used to extract values of the strong coupling strength α_S and the top quark pole mass (m^pole_t) using several sets of parton distribution functions (PDFs). The measurement of m^pole_t exploits the sensitivity of the tt^- invariant mass distribution to mpolet near the production threshold. Furthermore, a simultaneous fit of the PDFs, α_S, and m^pole_t is performed at NLO, demonstrating that the new data have significant impact on the gluon PDF, and at the same time allow an accurate determination of α_S and m^pole_t. The values α_S(m_Z) = 0.1135^+0.0021_-0.0017 and m^pole_t = 170.5 ± 0.8 GeV are extracted, which account for experimental and theoretical uncertainties, the latter being estimated from NLO scale variations. Possible effects from Coulomb and soft-gluon resummation near the tt^- production threshold are neglected in these parameter extractions. A rough estimate of these effects indicates an expected correction of m^pole_t of the order of +1 GeV, which can be regarded as additional theoretical uncertainty in the current m^pole_t extraction.