We present a systematic analysis of the extended X-ray emission discovered around 35 FR II radio galaxies from the revised Third Cambridge Catalog (3CR) Chandra Snapshot Survey with redshifts between 0.05 and 0.9. We aimed to (i) test for the presence of extended X-ray emission around FR II radio galaxies, (ii) investigate whether the extended emission origin is due to inverse Compton (IC) scattering of seed photons arising from the cosmic microwave background (CMB) or thermal emission from an intracluster medium (ICM), and (iii) test the impact of this extended emission on hot-spot detection. We investigated the nature of the extended X-ray emission by studying its morphology and compared our results with low-frequency radio observations (i.e., ∼150 MHz) in the TGSS and LOFAR archives, as well as with optical images from Pan-STARRS. In addition, we optimized a search for X-ray counterparts of hot spots in 3CR FR II radio galaxies. We found statistically significant extended emission (>3σ confidence level) along the radio axis of ∼90% and in the perpendicular direction of ∼60% of the galaxies in our sample. We confirmed the detection of seven hot spots in the 0.5-3 keV energy range. In the cases where the emission in the direction perpendicular to the radio axis is comparable to that along the radio axis, we suggest that the underlying radiative process is thermal emission from the ICM. Otherwise, the dominant radiative process is likely nonthermal IC/CMB emission from lobes. We found that nonthermal IC/CMB is the dominant process in ∼70% of the sources in our sample, while thermal emission from the ICM dominates in ∼15% of them.