1) “He sat there [silent / silently].”
Here I understand both are acceptable with the difference that the former describes ‘his state of being while he sat’ and the latter does ‘the way he sat’.
2) Then, which is correct in “We sat on the picnic mat [peaceful / peacefully].”?
I think the parse of the sentence with ‘sat on’ is different with the one with ‘sat’, since ‘sat on’ as an intransitive action verb implies “moved from a standing position to a sitting position on the picnic mat”.
Thus, while I think “peacefully“ - ‘in a quiet and calm way’ only would be fit in 2) above, I would appreciate on your responses.