  We present a high sensitivity spectral line survey of CO(1-0), CO(2-1),
CO(3-2) and [CI](1-0) in 40 local (ultra) luminous infrared galaxies
((U)LIRGs), all with previous Herschel OH119 $\mu$m observations. We use
single-dish observations (PI and archival) conducted with APEX, complemented
with ALMA and ACA data. We study the total emission and pay special attention
to the extended low-surface brightness components. We find a tight correlation
between low-J CO and [CI] line luminosities suggesting their emission arise
from similar regions, at least when averaged over galactic scales. We estimate
a median CO-to-H$_2$ conversion factor of $1.7\pm 0.5$ M$_{\odot}$ (K km
s$^{-1}$ pc$^2)^{-1}$ for ULIRGs, using [CI] as an independent tracer. We
derive median galaxy-integrated CO line ratios ($r_{21}$, $r_{31}$ and
$r_{32}$), as well as $r_{CICO}$, significantly higher than normal star forming
galaxies, confirming the exceptional molecular gas properties of ULIRGs. We
find that $r_{21}$ and $r_{32}$ are poor tracers of CO excitation in ULIRGs,
while $r_{31}$ shows a positive trend with $L_{IR}$ and SFR, and a negative
trend with the H$_2$ gas depletion timescales ($\tau_{dep}$). When studying CO
line ratios as a function of gas kinematics, we find a positive relation
between $r_{21}$ and $\sigma_v$, which can be explained by CO opacity effects.
We find that the linewidths of [CI] lines are ~10% narrower than CO lines,
which may suggest that the low optical depth of [CI] can challenge its
detection in diffuse, low-surface brightness outflows, and so its use as a
tracer of CO-dark H$_2$ gas in these components. Finally, we find that higher
$L_{AGN}$ are associated to longer $\tau_{dep}$, consistent with the hypothesis
that AGN feedback may reduce the efficiency of star formation.
