The gear card 'missile' says I can discard one success and exhaust a card to deal damage. Do I have to play an attack card to use this or can I carry out any action?
No, use any action during your activation. I believe you are referring to the handgun which gives you that window after your activation. After activation means once you are done not anytime after you start...this has caused some confusion. See this thread to understand the flow of the turn.