The game is coming together really nicely, great job! I am going to try to answer these questions in as much depth as I can. This is not a criticism of your game, just a way to try to be helpful.
1. I thought the player movement was intuitive and smooth. Having added movement capabilities could be fun but it is something you might have to play around with to see what works and what doesn't. If you have to collect eggs to rescue them it could be fun to have to go get them instead of shooting projectiles at them. Otherwise, there isn't a big need to move up/down since the birds are fairly easy to hit and you can only shoot up. Dashing might be something fun to try to implement, especially if the player is invulnerable while dashing and can only use it every so often. This could add some strategy in to help avoid the dive-bombing enemies.
2. I think the projectiles that the player shoots at the enemies are fine. They are a little slow to watch but the easily hit oncoming enemies so it doesn't hinder the player. If you increase enemy movement/speed then I think increasing the speed of the projectiles will be necessary too. It would be cool to see a powerup for the player that would allow the projectiles to either move faster or to shoot multiple directions at the same time. You could try having the powerups come as a collectible falling down the screen for the player to collect or you can have it be a special key/mouse button.
3. Since there isn't a way to regenerate health or pick up items that give health, I think the player has a fairly good amount of health. The amount of damage the enemies give is a good amount too. I think if the enemies were a little faster/unpredictable it would add to the difficulty of the game. So, if you make some small changes to the enemies without changing the health it'll make the game harder. Right now, it is fairly easy to go through the waves of enemies without losing too much health. The only exception is when the enemies dive bomb the player at the start of a wave. They do a lot of damage which doesn't really feel fair since the player doesn't have much time to react. Having a delay when the wave starts might help with that. In the middle of a level, the dive-bombing birds are fun to try to avoid since they are so quick.
4. Overall, I think having more movement from the enemies would increase the difficulty and make it more fun. It is really fun when the dive bomb the player and move around but right now most of the birds just stay in one place which makes them easy to hit with the projectiles. So, I would love to see more overall movement from the enemies. I would also love to see the baby birds/eggs that the player is trying to find since right now they are missing from the game.
Also, this is just a side-note: I have an ultra-wide monitor and the game plays full-screen by default. This made a lot of the text very blurry. When I played it on a smaller monitor it didn't adjust properly so the panel on the left with the health was cut off a bit.
Overall, great job! I love the art style. It really communicates the feel and story that your team pitched. The background scrolling in the background is smooth and it just makes the game come across as really fun/cute. With a few tweaks to the enemy behaviors, I think the game will end up being really fun to play. I look forward to the final product.
-Breanna Flickinger