  We present a comprehensive study of the physical origin of radio emission in
optical quasars at redshifts z < 2.5. We focus particularly on the associations
between compact radio emission, dust reddening, and outflows identified in our
earlier work. Leveraging the deepest low-frequency radio data available to date
(LoTSS Deep DR1), we achieve radio detection fractions of up to 94%,
demonstrating the virtual ubiquity of radio emission in quasars, and a
continuous distribution in radio loudness. Through our analysis of radio
properties, combined with spectral energy distribution modeling of
multiwavelength photometry, we establish that the primary source of radio
emission in quasars is the AGN, rather than star formation. Modeling the dust
reddening of the accretion disk emission shows a continuous increase in radio
detection in quasars as a function of the reddening parameter E(B-V),
suggesting a causal link between radio emission and dust reddening. Confirming
previous findings, we observe that the radio excess in red quasars is most
pronounced for sources with compact radio morphologies and intermediate radio
loudness. We find a significant increase in [Oiii] and Civ outflow velocities
for red quasars not seen in our control sample, with particularly powerful
[Oiii] winds in those around the radio-quiet/radio-loud threshold. Based on the
combined characterisation of radio, reddening, and wind properties in our
sample, we favor a model in which the compact radio emission observed in
quasars originates in compact radio jets and their interaction with a dusty,
circumnuclear environment. Our results align with the theory that jet-induced
winds and shocks resulting from this interaction are the origin of the enhanced
radio emission in red quasars. Further investigation of this model is crucial
for advancing our understanding of quasar feedback mechanisms and their role in
galaxy evolution.
