[DataNoh]

For example, say I fail Cardassian Processing and my opponent has no crew/Away Team present. Does the dilemma get used as a trap card for the captive? Does it go back under the mission to be encountered again? Or does it have to stay out as the trap card?

[Commander Joe]

From the Dilemma Resolution Guide.

CARDASSIAN PROCESSING [S/P]
Unless Diplomacy, Honor, Transporter Skill, and CUNNING>30 OR Law, SECURITY, and STRENGTH>40, one personnel is captured (random
selection).

If Away Team/crew meets conditions (Diplomacy, Honor, Transporter Skill, and CUNNING>30 OR Law, Security and STRENGTH>40), discard dilemma; mission continues.

Otherwise, randomly select a personnel to be captured. If present, opponent's crew or Away Team at same location takes custody of captive. Away Team/ship and crew are stopped; replace dilemma under mission to be encountered again.

Otherwise, place dilemma on captive as a “trap” card and place captive on opponent's side of the spaceline (discard dilemma if opponent takes custody of captive). Away Team/ship and crew are stopped.

[DataNoh]

Wow, can't believe I didn't look at that. Thank you for pointing out the obvious to me!!

[Takket]

For purposes of a card like Stratagem, is a personnel “opponent’s captive” if they are only held by a trap card? Or does someone not actually become a captive until they are “taken into custody”?

[DataNoh]

Personnel held by a trap card should be considered captives. The glossary says a trap card is "a marker of captive status".