Your question is a bit fuzzy. I will do my best to answer and let you complain about what I did not understand properly. The references that I give are just papers that I know and like. There is much more to read on the topic.
I will use the following sign convention for $\lambda$,
$$ \partial_t h + \frac{\lambda}{2}\left(\vec{\nabla}h\right)^2 = \nu \nabla^2 h + \eta \, . \qquad (1) $$
I assume that you know what every term stands for. All the parameters are positive and $\eta(t,\vec{x})$ is a stochastic noise.
Galilée invariance: KPZ equation is invariant under Galilei transformations. If $h(t,\vec{x})$ is a solution of Eq. (1), then
$$\tilde{h}(t,\vec{x}) = h(t,\vec{x}-t \vec{u}) + \frac{1}{\lambda} \vec{u} \cdot \tilde{x} - \frac{u^2 t}{2 \lambda} \, , \qquad (2)$$
is a solution as well (with $\tilde{\eta}(t,\vec{x}) = \eta(t,\vec{x}-\vec{u}t)$). Note that the last term is quadratic in $\vec{u}$ and is usually not considered since only infinitesimally small values of $\vec{u}$ are needed for the Ward identities. Note that both sides of Eq. (1) are invariant under such a transformation independently. This implies that any function of (in particular) the left-hand-side will be a Galilei scalar.
This transformation is called Galilei invariance because Eq. (1) can be mapped onto Burgers equation,
$$ \partial_t \vec{v} + \left(\vec{v} \cdot \vec{\nabla}\right) \vec{v} = \nu \Delta \vec{v} + \vec{\nabla}\eta \, , \qquad (3)$$
with the definition, $\vec{v} = \lambda \vec{\nabla}h$. Burgers' equation is very similar to Navier-Stokes and describes some kind of hydrodynamic velocity field. The transformation (2) applied to $\vec{v}$ becomes
$$ \tilde{\vec{v}}(t,\vec{x}) = \vec{v}(t,\vec{x}-t\vec{u}) + \vec{u} \, .$$
It is a Galilei transformation.
Scale invariance: There is no easy was to see that Eq. (1) is scale invariant. It takes a lot of work to see it. This can be done using Renormalisation Group (RG) techniques (see e.g. this and more references in it's bibliography) numerical simulations or even experiments. Note that the $d=1$ case has been solved analytically and the scale invariance of Eq. (1) proven mathematically. See e.g. this. Often it is simply just assumed that the stationary state is scale invariant.
In particular, scale invariance of the steady state implies that there are two scaling exponents ($\chi$ and $z$) and a scaling function $g(x)$ such that
$$ \langle \left[h(t+\tau,\vec{x}+\vec{r})-h(t,\vec{x}) \right]^2 \rangle = r^{2\chi} \, g\left(\frac{\tau}{r^z}\right) \, .$$
You can write similar expressions for all the correlation functions. Note that the steady state is invariant under space time translations as well as space rotations, $\left|\vec{r}\right| = r$.
For $d=1$, $\chi$ and $z$ are known. Huge amounts of work go into computing them for higher dimensions. Galilei invariance is a big help here because it relates one exponent to the other,
$$ \chi + z = 2 \, . \qquad (4)$$
Indeed, Galilei invariance insures (in particular) that $\lambda$ is not changed by the renormalisation group. Whatever the level of coarse graining, the two terms of the left-hand-side of Eq. (1) will always be related in the same way. If your system is scale invariant you can rescale space, time and field as
$$ x = \kappa \hat{x} \, , \quad t = \kappa^z \hat{t} \, , \quad h = \kappa^\chi \hat{h} \, .$$
Then the left-hand-side of Eq. (1) is rescaled as
$$ \kappa^{\chi-z} \partial_{\hat{t}} \hat{h} + \kappa^{2\chi-2} \left(\hat{\nabla}\hat{h}\right)^2 = \kappa^{\chi-z} \left[ \partial_{\hat{t}} \hat{h} + \kappa^{\chi+z-2} \left(\hat{\nabla}\hat{h}\right)^2 \right] \, .$$
The whole thing does not change (up to a prefactor that is absorbed into the renormalisation of $\nu$ and $\eta$) only if Eq. (4) is fulfilled.
Advective non-linearity: The left-hand-side of Eq. (1) should be considered as a block because in the hydrodynamic representation (Eq. (3)), it is the advective time derivative,
$$\frac{d \vec{v}(t,\vec{x}(t))}{d t} = \partial_t \vec{v} + \left(\vec{v}\cdot \vec{\nabla}\right) \vec{v} \, .$$
This shows as well why it is Galilei invariance that is responsible for the conservation (under RG transformations) of $\lambda$.
Ward identities: You can find the Ward identities as well as their derivation here.
