Proving membership in W-hierarchy when problem is not parameterized by its solution size
The answer to your updated Question (1) &quot;When a problem is parameterized by something other than the size of a solution (and as a result, the size of a solution is still unbounded in terms of the parameter), then [is it] W[t]-hard for all t, at least?&quot; is no. Independent Set for example is FPT with respect to the treewidth $\omega$ of the input graph. The solution size $k$ (essentially the size of a largest independent set of the input graph) can be much larger than $\omega$.
In general, a parameter can be anything that is related to the input instance. The standard parameter solution size is just one of many different ways to parameterize a problem.