If my opponent has an Abomasnow with "Freezing Disaster" Ability in play, can I use Munkidori's "Adrena-Brain" Ability to move damage counters off one of my Pokemon or would that count as "healing"?
Yes, you can do that. Moving damage counters off a Pokemon is not the same as healing damage (similar as to how moving damage counters onto a Pokemon is not the same as doing damage to it).
Source: TPCi Rules Team (2024-06-27)
Can I use Dedenne ex's "Tail Swap" attack to move damage counters around if Abomasnow's "Freezing Disaster" Ability is in play?
Yes you can, since "moving" damage counters is not the same as "healing" damage.
Source: Paldea Evolved FAQ; TPCi Rules Team (2023-06-08)