An analytical model of open-cavity second sound resonators is presented and validated against simulations and experiments in superfluid helium using a new resonator design that achieves unprecedented resolution. The model incorporates diffraction, geometrical misalignments, and flow through the cavity, and is validated using cavities with aspect ratios close to unity, operated up to their 20th resonance in superfluid helium. An important finding of this study is that resonators can be optimized to selectively sense either the quantum vortex density carried by the throughflow -as typically done in the literature- or the mean velocity of the throughflow. We propose two velocity probing methods: one that takes advantage of geometrical misalignments between the tweezers plates, and another that drives the resonator non-linearly, beyond a threshold that results in the self-sustainment of a vortex tangle within the cavity. A new mathematical treatment of the resonant signal is proposed to adequately filter out parasitic signals, such as temperature and pressure drift, and accurately separate the quantum vorticity signal. This elliptic method consists in a geometrical projection of the resonance in the inverse complex plane. Its effectiveness is demonstrated over a wide range of operating conditions. The resonator model and elliptic method are being utilized to characterize a new design of second-sound resonator with high resolution thanks to miniaturization and design optimization. These second-sound tweezers are capable of providing time-space resolved information similar to classical local probes in turbulence, down to sub-millimeter and sub-millisecond scales. The principle, design, and micro-fabrication of second sound tweezers are being presented in detail, along with their potential for exploring quantum turbulence.