Interesting game. I had a few difficulties playing it due to the turning mechanics. Considering there's quite a bit of platforming in this game, it's vital the character can turn properly and reorient in tight spaces. This is probably a pivot point/transform offset issue. If your player is a prefab, you might what to check your player (parent object + any children) are all set the transform to (0, 0, 0) inside the prefab. It could also be a center of mass issue. If you have rigidbody-based movement, unity calculates center of mass based on colliders. If you have multiple colliders on your player (like for sensor detection), the rigidbody could be trying to rotate around the weighted center of all those shapes combined. It also might be a code issue. If you used transform.RotateAround() with a target point offset from the player's center, it would cause the character to travel in that arching turn.
The player also seemed to vibrate when moving along the ground. In the past, I've found that if the player's position is being updated in FixedUpdate but the camera is following in Update, it can cause a jittery look because the camera is trying to find a player position that hasn't been calculated yet. But I didn't notice the jittering while jumping, so it may be a floor snag issue with the physics engine. Or if you're using a character controller and you have some slightly buggy ground check logic, it could cause a grounded/not grounded flickering. You also mentioned having problems with a floating character. If you have some 'stick to ground' code running every frame while the player's on the ground, it might be fighting the rigidbody's physics or jump arc. If you add a physics material to your player with friction set to 0, and the jittering stops, then it's a friction snagging issue. Otherwise it might be code related.
But I really like your textures, level design, and story concept. The roving enemies were cool too. Nice job!