On my first run through I didn't even really notice the green "ship health" bar decreasing because I was more concerned with my player's health bar (Which was decreasing much quicker). I think both of the things you mentioned would be good (clarity on aliens damaging the ship & combining the two actions).
Thank you so much for saying that. That could mean that one of the games major flaws stems from the fact that its not clear to the player that they have to keep both themselves and the ship alive, and also that the ship takes damage. What you noticed that felt disjointed could be because of any number of things, but that could be a good place to start. Thank you again for your feedback!