  We present high-resolution 1.5 $-$ 6 GHz Karl G. Jansky Very Large Array
(VLA) and Hubble Space Telescope (HST) optical and infrared observations of the
extremely active repeating fast radio burst (FRB) FRB 20201124A and its barred
spiral host galaxy. We constrain the location and morphology of star formation
in the host and search for a persistent radio source (PRS) coincident with FRB
20201124A. We resolve the morphology of the radio emission across all frequency
bands and measure a star formation rate SFR $\approx 8.9\,M_{\odot}$ yr$^{-1}$,
approximately $\approx 2.5-6$ times larger than optically-inferred SFRs,
demonstrating dust-obscured star formation throughout the host. Compared to a
sample of all known FRB hosts with radio emission, the host of FRB 20201124A
has the most significantly obscured star formation. While HST observations show
the FRB to be offset from the bar or spiral arms, the radio emission extends to
the FRB location. We propose that the FRB progenitor could have formed in situ
(e.g., a magnetar born from a massive star explosion). It is still plausible,
although less likely, that the progenitor of FRB 20201124A migrated from the
central bar of the host. We further place a limit on the luminosity of a
putative PRS at the FRB position of $L_{\rm 6.0 \ GHz}$ $\lesssim$ 1.8 $\times
10^{27}$ erg s$^{-1}$ Hz$^{-1}$, among the deepest PRS luminosity limits to
date. However, this limit is still broadly consistent with both magnetar
nebulae and hypernebulae models assuming a constant energy injection rate of
the magnetar and an age of $\gtrsim 10^{5}$ yr in each model, respectively.
