(This is Henry)
1. I didn't feel like I was missing a different movement ability while playing, but I think it would be fine to add one as well. I think the answer to that question really depends on what you're going for. Maybe that could tie in somehow to my answer for the fourth question.
2. I'm not sure what projectiles you're referring to, as the only projectiles I saw were the ones coming from the bird (maybe that answers your question, haha). If you do mean the bird projectiles, their travel speed was fine, and spamming them was also fine. If you're referring to the dive bombing birds with this question, they were very hard to evade when they dive bombed the player right at the start of a stage, and I think they could be slowed down a bit.
3. I made it to level 2, stage 11/3, so I'm assuming I would have passed that level with ease. The only thing I ended up taking damage from were the dive bombing birds, which, as I mentioned above, felt a bit fast and cheap. Pretty much all the damage I ever took, at least from what I saw, was at the start of a stage. I think the player could have less health, especially if the speed of those birds is tweaked.
4. I'd like two things related to shooting. While playing I ended up just mashing the left mouse button, so I think I'd prefer to be able to hold the button down to save my poor finger the effort. That being said, I also think it would be good for the player to have some reason to not be shooting at some point- maybe they charge up power to do more damage on the next bullet when not attacking or are able to move faster. Even if you don't end up implementing a "hold the button" solution I think the game needs this, as the optimal strategy for the game is to just shoot constantly. If there was some sort of penalty already in place, I didn't notice it while playing.