Something-Treewidth Property
For question $1$: any bidimensional parameter has this property on general graphs. A parameter $s(G)$ is bidimensional if the value of $s(G) \geq s(H)$ for every minor $H$ of $G$, and if $s$ is ``large'' on grids.
In applications to PTASes, sub exponential algorithms and kernels on minor-free classes of graphs, &quot;large&quot; means that there exists a constant $c$, such that the value of $s$ on a $t$ times $t$ grid is at least $ct^2$. This is what you most likely will find if you do a google search for ``bidimensionality''
However, for your question it is sufficient that $s$ grows to infinity on $t$ times $t$ grids as $t$ grows to infinity. This is because any graph with large enough treewidth will contain a large enough grid minor. So, to conclude, if $s$:

is closed under minors
is arbitrarily large on $t$ times $t$ grids for large enough $t$

Then s has the s-treewidth property. See the recent parameterized complexity book ( http://parameterized-algorithms.mimuw.edu.pl ) in the treewidth chapter for more info.