The definitive source for lookup of Picard tag field names is https://picard.musicbrainz.org/docs/mappings/
From there it becomes clear, that what you refer to as bug2 above is intended behavior: what's displayed as "MusicBrainz Recording Id" in Picard is in fact written as UFID://musicbrainz.org (or MUSICBRAINZ_TRACKID for non-ID3v2 tagging formats).
In Mp3tag it's simply MUSICBRAINZ_TRACKID as listed in the overview table at Add mappings for more MusicBrainz tags
Let me know if I'm missing something, so far I don't see an issue.