  ($ABRIDGED$) We probe the physical properties and large-scale environment of
radio AGN in the faintest FR population to-date, and link them to their radio
structure. We use the VLA-COSMOS Large Project at 3 GHz, with resolution and
sensitivity of 0".75 and 2.3 $\mu$Jy/beam, respectively, to explore the FR
dichotomy down to $\mu$Jy levels. We classify objects as FRIs, FRIIs or hybrid
FRI/FRII based on the surface-brightness distribution along their radio
structure. Our control sample is the jet-less/compact radio AGN (COM AGN) which
show excess radio emission at 3 GHz VLA-COSMOS exceeding what is coming from
star-formation alone; this sample excludes FRs. Largest angular projected sizes
of FR objects are measured by a machine-learning algorithm and also by hand,
following a parametric approach to the FR classification. Eddington ratios are
calculated using scaling relations from the X-rays, while we include the jet
power by using radio luminosity as a probe. We investigate their host
properties (star-formation ratio, stellar mass, morphology), and we explore
their incidence within X-ray galaxy groups in COSMOS, as well as in the density
fields and cosmic-web probes in COSMOS. Our sample is composed of 59 FRIIs, 32
FRI/FRIIs, 39 FRIs, and 1818 COM AGN at 0.03 $\le z \le$ 6. FR objects have on
average similar radio luminosities ($L_{\rm 3~GHz}\rm \sim
10^{23}~W~Hz^{-1}~sr^{-1}$), spanning a range of $\rm
10^{21-26}~W~Hz^{-1}~sr^{-1}$, and lie at a median redshift of $z ~\sim ~1$.
FRs reside in their majority in massive quenched hosts ($M_{*}~> 10^{10.5}
M_{\odot}$), with older episodes of star-formation linked to lower X-ray galaxy
group temperatures, suggesting radio-mode AGN quenching. Irrespective of their
radio structure, FRs and COM AGN are found in all types and density
environments (group or cluster, filaments, field).
