Regarding tame covers: For log-schemes there are the notions of log-étale and log-smooth morphisms, which behave very similarly to the classical notions of étaleness and smoothness.

If $X\subset \overline{X}$ and $Y\subset \overline{Y}$ are open immersions of smooth $k$-schemes, for, say, $k$ a field, such that the complements of $X$ and $Y$ are strict normal crossings divisors, then $\overline{X}$ and $\overline{Y}$ get canonical (fine, saturated) log-structures. Lets call the log-schemes $X^{\log}$ and $Y^{\log}$. If $f:X\rightarrow Y$ is a finite étale morphism, extending to a finite morphism $\bar{f}:\overline{X}\rightarrow \overline{Y}$, then $f$ induces a morphism of log-schemes $f^{\log}:X^{\log}\rightarrow Y^{\log}$, and $f^{\log}$ is log-étale if and only if $\bar{f}$ is a tame covering in the usual sense. So it "behaves" like an étale covering in the category of log-schemes. For example, one can develop a theory of log-fundamengal groups and so on. A very nice reference for this is Jakob Stix thesis, which can be found on his homepage.

In fact log-étaleness is more general: For $f^{\log}$ to be log-étale, $\bar{f}$ does not have to be a finite morphism; certain non-finite ones are allowed, for example so called "log-blowups". As far as I understand they play a crucial role in developing log-étale cohomology. A good reference for this and much much more is

Illusie, Luc. An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology. (English summary) Cohomologies p-adiques et applications arithmétiques, II. Astérisque No. 279 (2002), 271–322.