If I use Steenee's "Captivate" attack while Spikemuth stadium is in play, does the opponent put any damage counters on their Pokemon?
No, Captivate would not trigger Spikemuth since it was not moved during the opponent's turn.
Source: Darkness Ablaze FAQ (2020-07-31), TPCi Rules Team (2020-08-13)

Relevant cards

Note: All applicable cards may not be listed below.
SWSH3_15
SWSH3_15
SWSH3_170
SWSH3_170