Does this mean physically or structurally speaking in an interesting way, or that he says interesting things?
Without any context, both are a possibility really, but rather latter that the former, I think.
I am unclear as to why ending is -ie in ciekawie. "Ciekawie" is an adverb, here, correct? As an adverb, will it always be "ciekawie" or does it have something to do with the subject "on"?
Hey! I bring good news! I imagine you will love it in fact: Adverbs do NOT decline in Polish, so they only have one form – one thing that's easy in Polish. ;)
What would be the translation of "he is always talking very interestingly"?
"On zawsze mówi bardzo [ciekawie/interesująco]".
Or "On zawsze mówi w bardzo [ciekawy/interesujący] sposób", although that's technically "in a very interesting [way/manner]".
I am inclined to say in an intersting manner. The disparity between your and mine english is my problem at times.