Hairer in his [spdes notes][1] on pg.6, says that GFF is the stationary solution of $u_{t}(z)=\Delta u(z)+\xi(z,t)$, where 
$\xi$ is the space-time white noise
$$\xi(x,t)=\sum \sqrt{\lambda_{k}} B_{k}(t)e_{k}(x)$$
for iid Brownian motions $B_{k}$ and L2 basis $e_{k}$. So that means we take
$$u(x,t)=\Delta^{-1}\xi(x,t). $$
and that at each fixed time it is a GFF 
$$h(x)=\sum \frac{1}{\sqrt{\lambda_{k}}} B_{k}(t)e_{k}(x).$$

Does SHE converge to a steady state? Is it the GFF? Do we also have some limiting results? Any rates?

From the same notes if we compute and expand the covariance for SHE we have

$$E(u(t,x)u(t,y))=log(\frac{1}{|x-y|^{2}})+log(t)-\gamma+O(\frac{|x-y|^{2}}{t})$$

and so even though for each fixed we have a GFF like object, for $t\to \infty$ the covariance becomes infinite.   

[1]: http://www.hairer.org/notes/SPDEs.pdf