Resummation phenomenology and PDF determination for precision QCD at the LHC

With the ongoing Run 3 of the LHC and its upcoming High-Luminosity upgrade, there is a growing need to study observables with high precision both experimentally and theoretically. To increase precision on the theory side, improvements of fixed-order perturbative predictions, resummation of logarithmic enhancements and accurate determination of proton structure are required. This thesis explores the latter two topics. We discuss high-energy logarithms and their resummation techniques, introducing an extension of the HELL formalism for multi-differential distributions in transverse momentum, rapidity and invariant mass. We apply this framework to heavy-quark pair production at the LHC, studying the kinematics of both a single quark and the final-state pair. An additional discussion is dedicated to a possible extension of the kt-factorisation framework, which underlies high-energy resummation, to capture next-to-leading logarithmic corrections. To test this hypothesis, we delve into the computation of a NLO off-shell coefficient function using Higgs-induced DIS in the infinite top mass limit as a benchmark process and report a partial result. Beside high-energy logarithms, we consider the determination of transverse-momentum distributions from a high mass system with additional QCD radiation and exclusive production cuts. Specifically, we focus on $HW^+$ production with a jet veto and analyse the Higgs transverse momentum spectrum at NNLO, using qt-subtraction. We complement the fixed order study with NNLL resummation of jet-veto logarithms and linear power correction in ptHW using the RadISH formalism. Finally, in the last project pertaining to this thesis we consider the problem of Parton Distribution Function determination. We propose a minimal parametrisation guided by physical arguments and investigate its performance in fitting the HERA dataset with NLO QCD theory predictions.