Unfortunately I haven't figured out exactly what is causing the bouncing. I will say that behind the scenes the player has a few platformer quality of life things like  variable jump height (tapping jump gives a small hop, holding jump gives a big jump), coyote time (small window of time after you leave the ground where you can still jump), and a jump buffer (pressing jump when you are not on the ground preserves that jump input for a short time and has you jump if you touch the ground while it's preserved). That said, I think the issue has to do with the jump buffer and/or how I'm getting jump press inputs because it is trying to jump whenever it touches the ground.

When it comes to the background images, I knew I wanted them to be big enough that the player would only be able to see a portion of it based on where they were on the stage. So if you were on the far left of the stage, you would only see the far left of the background and as you went right you would start seeing it move with you until you reached the right edge.
Originally, I thought to just take the backgrounds my artist gave and scale them up to fill the entire play area, but scaling them that much made it so big it was difficult tell what you were even looking at. So a much better solution that I ended up using in game was to have the background images scaled to be a little bigger than the lens of the camera, and have them follow the camera with an offset (parallax).

Sorry if it was unclear. When you defeat enemies, they drop "souls" which you spend to open those boxes (you can see how many souls you have in the top left). The boxes give you items that improve your character and make it easier to get around and/or fight enemies. The game was balanced around you getting at least a few items per stage rather than strictly focusing on the ship.