Liquid Argon Time Projection Chamber (LArTPC) neutrino detectors have emerged as a detector technology poised to perform measurements of neutrino interactions with unprecedented precision and as a result answer some of the largest open questions in neutrino physics in the coming decade. This thesis describes methods developed to tackle the computational challenges faced as LArTPC detectors increase in scale and complexity moving towards the multi-kiloton DUNE detectors. These include a new, approximated, model that enables rapid simulation of scintillation light in very large scale detectors, as well as the first demonstration of running the LArSoft software framework on a high performance computer. The high precision of LArTPC detectors designed for next-generation neutrino measurements enables them to also search for Beyond the Standard Model physics produced in high-energy proton--fixed-target collisions in neutrino beams. This thesis presents searches for two dark-sector models performed with the ArgoNeuT experiment: Heavy Neutral Leptons and Heavy QCD Axions. Between them, these models can provide solutions for various unresolved puzzles including neutrino mass generation, the baryon asymmetry of the universe, dark matter and the strong CP problem. In both cases, the dark-sector particles could be produced in the NuMI neutrino beam and can then decay to a pair of oppositely charged muons observable in ArgoNeuT and the downstream MINOS near detector. Both measurements required the development of novel experimental selection techniques and enabled new constraints to be set on the existence of these particles in previously unexplored parameter-space. These searches are both the first of their kind in LArTPC neutrino detectors and pave the way for searches at future neutrino facilities.