Can I use Sitrus Berry to heal damage received from Poison/Burn in between turns?
No, you cannot. Sitrus Berry only works at the end of a turn.
Source: TPCi Rules Team (2020-02-06)
If an attack would KO a Pokemon, can you use Sitrus Berry to heal it first?
No. KO's have to be resolved before the end of the turn, and Sitrus Berry heals at the end of the turn after KO's have been evaluated.
Source: TPCi Rules Team (2020-02-06)
The wording of Sitrus Berry is now as follows, "At the end of each turn, if the Pokémon this card is attached to has 3 or more damage counters on it, heal 30 damage from it and discard this card."
Source: TPCi Announcement (2020-01-09), TPCi Rules Team (2020-01-09)