I'm glad you liked it! :)
1.
I used KinematicBody2D for the player and Static. for the platforms and floors / walls.
I didn't notice the bouncing but I think I know what can be the problem. I used capsule as the CollisionPolygon for the player. I noticed that the rectangle as bound of player doesn't work so well with diagonal tiles, I can't climb up for example. I think I know how I could I fix that; adding a second collision as foot, I remember seeing something like that in a FPS example project.
2.
The "repeating-jump" when I keep pushing the SPACE wasn't irritating for me...however after reading your comment I see why it can be a problem.
I think I could add a key for turning repeating-jump on / off.
After the voting period I'll update it, thanks :)