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It often looks like you don't jump as early as you should in staircases, instead ending up caught on each step along the way. I'm assuming this is just the way it controls? I see there's some notes on stairs in the guide.
I'm sure you tested shooting more enemies to reduce lag.
I tried to jump as early as possible, but I'm sure there were some cases where my input was slightly off and I ended up landing on the side of the next step. Assuming that's what you meant.
The only enemies you can shoot are the bouncing ball enemies. The run would have looked very different if every enemy was destoryable. There are a couple of situations, where you have the option to shoot a ball enemy or not. The considerations in those cases are described in the guide.
It's not possible to jump down the pits in bonus 5, 6 and 7. It might look like it should be, but the platforms appear too early for it to work.