For missions where an enemy gains actions per hero activation (such as

is Jynn able to use her quick draw ability? Or because it is just an action, not an activation, is she unable to respond?

I would say she would not be able to. There are other game effects (such as the Elite Imperial Officer) that grant a move action or an attack action outside normal activation that Jyn would not be able to respond to, since her ability specifically states "Use at the start of a hostile figure's activation..."

I think it even says in the mission rules "instead of activating normally performs 1 action..." so I would also say no because said model doesn't activate at all.

I got a reply from FFG stating that these abilities can activate on each action performed by the model. Every action granted counts as start of activation or end of activation.