I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one.

1. For the algebraic cobordism theory $MGL$ and a smooth variety $X$ over a (perfect?) field is it true that $MGL^{2n+i,n}(X)=0$ for any $n\in \mathbb{Z},i>0$? More generally, are there any reasonable restrictions on a (oriented?) ring spectrum $E$ in $SH$ that ensure the vanishing of
   $MGL^{2n+i,n}(E)$. In particular, is this question related with some sort of effectivity for spectra?

2. It is well known that 'shifts and twists' of the spectra $\Sigma(X_+)$ generate $SH$, where $X$ runs through all smooth $k$-varieties. If the characteristic of $k$ is $0$, resolution of singularities yields that it suffices to consider only smooth projective varieties here. Now, what statements of this sort are known for $k$ of characteristic $p>0$? I suspect that that one can deduce a similar result for $SH\otimes \mathbb{Z}_{(l)}$ for any prime $l\neq p$, ffrom the Gabber's l'-alterations theorem. Is this true? If this is too difficult, can one prove a similar statement for the triangulated category of $MGL$-modules?

What are the best references for these questions?