Check out Chapter 10 of Mazza-Voevodsky-Weibel "Lecture notes on motivic cohomology", which discusses étale motivic cohomology. The answers to your questions can be found there:
2) yes: Immediately after Definition 10.1, you find the vanishing $H^{p,q}_L(X,\mathbb{Z})=0$ if $q<0$. This follows directly from the definition of $H^{p,q}_L$ as étale hypercohomology of complexes $\mathbb{Z}(q)$ which are trivial for $q<0$.
1) no: The statement for motivic cohomology is a consequence of the fact that the Nisnevich topology has finite cohomological dimension cohomological dimension (equal to the Krull dimension of the scheme). There is no such thing for étale cohomology, and so you should not actually expect 1) to be true for étale motivic cohomology. Theorem 10.2 gives you an isomorphism $H^{p,q}_L(X,\mathbb{Z}/n)\cong H^{p,q}_{ét}(X,\mu_n^{\otimes q})$ for $q\geq 0$, $p\in\mathbb{Z}$ and $n$ prime to the characteristic of the field $k$ over which $X$ is defined. To get a concrete example of this failure, I guess we can take $X=\operatorname{Spec}\mathbb{R}$, $n=2$. The étale cohomology in this case is cohomology of the Galois group $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$ with coefficients in $\mu_2$. This is 2-periodic (cohomology of a finite group) and $H^1_{ét}(X,\mu_2)=\mathbb{R}^\times/(\mathbb{R}^\times)^2\cong\mathbb{Z}/2$. So there are non-trivial cohomology groups in arbitrarily high dimensions.
