$L(\Gamma,G)$ is the algebra of endomorphisms of the representation $l^2( \Gamma/G)$ (with $\Gamma$ acting by left multiplication). This answer your second optional question. Also, by classical results on $W^*$-categories, the category of normal representations of $L(\Gamma,G)$ will be equivalent to the category of unitary representations of $G$ that are retract of sums of copies of $l^2(\Gamma /G)$, which generally allow to determine the type in concrete situations but I don't know what a general criterion would be. In fact, in the general case this algebra can also be of type $III$ which make me think there is no simple criterion.
In the special case where $G$ is finite, $l^2(\Gamma /G)$ is a retract of $l^2(\Gamma)$ hence $L(\Gamma,G)$ is a corner of $L(\Gamma)$. If in addition $\Gamma$ is ICC, then $L(\Gamma)$ is a factor and hence any non trivial corner will be Morita equivalent to $L(\Gamma)$. So $L(\Gamma,G)$ will be Morita equivalent to $L(\Gamma)$ and hence of the same type.
Also note that in this situation $L(\Gamma)$ and $L(\Gamma,G)$ will often be isomorphic, but by a "non natural" isomorphism which will not going to be compatible with the natural Morita equivalence...