My understanding is that sponsorship on at least the E-3 is difficult or impossible; that visa is not "immigrant intent", which ostensibly means something in the statute with respect to paths to permanent residence.
Just because a visa isn't dual intent doesn't mean you can't apply for a Green Card. TN, E-3 etc can all lead to a Green Card but you have to time it right.
Speaking as a successful E3->GC sponsoree I can tell you it’s not impossible (proof by counterexample FTW).
For L1 there's no issue.
Yeah - but vast majority of GC applications will be from H1-B pool