  With planned space-based and 3rd generation ground-based gravitational wave
detectors (LISA, Einstein Telescope, Cosmic Explorer), and proposed DeciHz
detectors (DECIGO, Big Bang Observer), it is timely to explore statistical
cosmological tests that can be employed with the forthcoming plethora of data,
$10^4-10^6$ mergers a year. We forecast the combination of the standard siren
measurement with the weak lensing of gravitational waves from binary mergers.
For 10 years of 3rd generation detector runtime, this joint analysis will
constrain the dark energy equation of state with marginalised $1\sigma$
uncertainties of $\sigma(w_0)$~0.005 and $\sigma(w_a)$~0.04. This is comparable
to or better than forecasts for future galaxy/intensity mapping surveys, and
better constraints are possible when combining these and other future probes
with gravitational waves. We find that combining mergers with and without an
electromagnetic counterpart helps break parameter degeneracies. Using DeciHz
detectors in the post-LISA era, we demonstrate for the first time how merging
binaries could achieve a precision on the sum of neutrino masses of
$\sigma(\Sigma m_{\nu})$~0.05 eV using $3\times10^6$ sources up to $z=3.5$ with
a distance uncertainty of $1\%$, and ~percent or sub-percent precision also on
curvature, dark energy, and other parameters, independently from other probes.
Finally, we demonstrate how the cosmology dependence in the redshift
distribution of mergers can be exploited to improve dark energy constraints if
the cosmic merger rate is known, instead of relying on measured distributions
as is standard in cosmology. In the coming decades gravitational waves will
become a formidable probe of both geometry and large scale structure.
