does the trap and mine damage support gem increase the damage of conversion trap minions?

couldnt find any info regarding the interaction between the two.

The support would work if minion uses traps, and there are only two types of specters that do.

The major snag here is conversion trap doesn't create a 'minion', it temporarily turns a monster into an 'ally' so no minion-related bonuses affect it. Actually, nothing you link with conversion trap will affect it, it isn't created using the skill like those dominating blow specters are.