  We study the scaling behavior and finite (physical) volume effects as well as
the Gribov copy dependence of the SU(2) Landau gauge gluon propagator on the
lattice. Our physical lattice sizes range from $(3.0 \mathrm{fm})^4$ to $(7.3
\mathrm{fm})^4$. Considering lattices with decreasing lattice spacing but fixed
physical volume we confirm (non-perturbative) multiplicative renormalizability
and the approach to the {\it continuum limit} for the renormalized gluon
propagator $D_{ren}(p)$ at momenta $|p| \ageq 0.6$ GeV >. The finite-volume
effects and Gribov copy influence turn out small in this region. On the
contrary, in the deeper infrared we found the Gribov copy influence strong and
finite-volume effects, which still require special attention. The gluon
propagator does not seem to be consistent with a simple pole-like behavior
$\sim (p^2+m_g^2)^{-1}$ for momenta $|p| \aleq 0.6$ GeV. Instead, a
Gaussian-type fit works very well in this region. From its width - for a
physical volume $(5.0 \mathrm{fm})^4$ - we estimate a corresponding infrared
(mass) scale to be $m_\mathrm{IR} \sim 0.7 $ GeV.
