Zostać means "stay" AND "become" ?
Yes. Sometimes it's good to use 'pozostać' to make it clear that you mean 'stay'.
Bo dotknął tamtą kobietę
"Bo dotknął tamtej kobiety". The verb "dotykać" is used with Genitive.
I think he did more than touch.
Isn't został "remained"?
"zostać" can technically mean "to remain", but rather only when it's the only interpretation that makes sense. Otherwise, use "pozostać", it's definitely "to remain".
Trying to get used to 'zostal' -not as 'stayed' but
Means both, depending on the context. In fact, I'd say that "became" may be the main meaning, as when you want things to be unambiguous and definitely mean "remained" or "stayed", that's "pozostał".
Is there a logic behind the instrumental being applied here? I would have thought accusative being more logical here
You could interpret this as the perfective version of być, which also requires instrumental.