  Using a set of high-resolution simulations we study the statistical
correlation of dark matter halo properties with the large-scale environment. We
consider halo populations split into four Cosmic Web (CW) elements: voids,
walls, filaments, and nodes. For the first time we present a study of CW
effects for halos covering six decades in mass: $10^{8}-10^{14}{h^{-1}{\rm
M}_{\odot}}$. We find that the fraction of halos living in various web
components is a strong function of mass, with the majority of
$M>10^{12}{h^{-1}{\rm M}_{\odot}}$ halos living in filaments and nodes. Low
mass halos are more equitably distributed in filaments, walls, and voids. For
halo density profiles and formation times we find a universal mass threshold of
$M_{th}\sim6\times10^{10}{h^{-1}{\rm M}_{\odot}}$ below which these properties
vary with environment. Here, filament halos have the steepest
concentration-mass relation, walls are close to the overall mean, and void
halos have the flattest relation. This amounts to $c_{200}$ for filament and
void halos that are respectively $14\%$ higher and $7\%$ lower than the mean at
$M=2\times10^8{h^{-1}{\rm M}_{\odot}}$, with low-mass node halos being most
likely splashed-back. We find double power-law fits that very well describe
$c(M)$ for the four environments in the whole probed mass range. A
complementary picture is found for the average formation times, with the
mass-formation time relations following trends shown for the concentrations:
the nodes halos being the oldest and void halo the youngest. The CW
environmental effect is much weaker when studying the halo spin and shapes. The
trends with halo mass is reversed: the small halos with $M<10^{10}{h^{-1}{\rm
M}_{\odot}}$ seem to be unaffected by the CW environment. Some weak trends are
visible for more massive void and walls halos, which, on average, are
characterized by lower spin and higher triaxiality parameters.
