FR-type radio sources at 3 GHz VLA-COSMOS: Relation to physical properties and large-scale environment

($ABRIDGED$) 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).