An analytical model of second-sound resonators with open-cavity is presented and validated against simulations and experiments in superfluid helium using a new design of resonators reaching unprecedented resolution. The model accounts for diffraction, geometrical misalignments and flow through the cavity. It is validated against simulations and experiments using cavities of aspect ratio of the order of unity operated up to their 20$^{th}$ resonance in superfluid helium. An important result is that resonators can be optimized to selectively sense the quantum vortex density carried by the throughflow -as customarily done in the literature- or alternatively to sense the mean velocity of this throughflow. Two velocity probing methods are proposed, one taking advantage of geometrical misalignements between the tweezers plates, and another one by driving the resonator non-linearly, beyond a threshold entailing the self-sustainment of a vortex tangle within the cavity. After reviewing several methods, a new mathematical treatment of the resonant signal is proposed, to properly separate the quantum vorticity from the parasitic signals arising for instance from temperature and pressure drift. This so-called \emph{elliptic method} consists in a geometrical projection of the resonance in the inverse complex plane. Its strength is illustrated over a broad range of operating conditions. The resonator model and the elliptic method are applied to characterize a new design of second-sound resonator of high resolution thanks to miniaturization and design optimization. When immersed in a superfluid flow, these so-called \emph{ second-sound tweezers} provide time-space resolved information like classical local probes in turbulence, here down to sub-millimeter and sub-millisecond scales. The principle, design and micro-fabrication of second sound tweezers are detailed, as well as their potential for the exploration of quantum turbulence