Does the short neuter form of adjectives always automatically constitute the corresponding adverb, or are there cases of the short neuter form and the adverb having different forms/stress patterns?
Равный yields the short neuter равнó but the adverb is рáвно. I think there are other adverbs that don't follow suit when the corresponding short adjective shifts stress to the last syllable.
Then, most passive participles with the suffix
-нн- lose the second
н in the short form, while the adverb keeps it geminated: растерянный — общество растеряно, but растерянно улыбнуться. That can also be accompanied by a non-matching stress shift: определенó (adj.) vs. определённо (adv.)