We numerically investigate the feasibility and limits of jointly estimating flow fields and unknown particle properties (e.g., position, size, and density) from Lagrangian particle tracking (LPT) data. LPT offers time-resolved, volumetric measurements of particle trajectories, which are markers of the carrier fluid motion. However, experimental tracks are spatially sparse and potentially noisy, and the problem of reconstructing flow fields may be further complicated by inertial particle transport, such that particle slip velocities must be determined to access the velocity field of the carrier fluid. To address this problem, we develop a data assimilation framework that couples an Eulerian representation of the flow with Lagrangian particle models, enabling the simultaneous inference of carrier fields and particle properties under the governing equations of disperse multiphase flow. We show that flow fields and particle properties can be jointly estimated in three representative regimes: (1) In a turbulent boundary layer with noisy tracer tracks (St to 0), flow fields and true particle positions are jointly estimated, which amounts to a physics-informed particle tracking problem; (2) in homogeneous isotropic turbulence seeded with inertial particles (St ~ 1-5), we demonstrate simultaneous recovery of flow states and particle diameters, showing the feasibility of implicit particle characterization; and (3) in a compressible, shock-dominated flow, we report the first joint reconstructions of velocity, pressure, density, and inertial particle properties (diameter and density), highlighting both the potential and certain limits of joint estimation in supersonic regimes. A systematic sensitivity study reveals how the seeding density, noise level, and Stokes number govern reconstruction accuracy for our method.