i'm not sure exactly why you got the game over loop there. There are a few things extra you could do to try to prevent the game over loop even when she does climb up the waterfall. I need to play through that section again and determine exactly what could prevent it....don't have a 100% consistent method for preventing that loop, but it's close.
don't really have a solution for the bird cage part....i wouldn't worry about wallrunning up to that pole though. It's not worth it if there's a chance to trigger the cutscene.
The double rewind bug happens when you use a rewind the instant the prince touches the ground....at least i think this is the cause. You're probably getting it on the 3rd rewind, correct? That's the one where the prince hits the ground. My suggestion to prevent this is to wait just a little bit longer after the prince hits the ground (so use the rewind about half a second later than you're currently using it). Practice this and see if you can still pull off the trick consistently, and also if it prevents the double rewind bug.
don't really have a solution for the bird cage part....i wouldn't worry about wallrunning up to that pole though. It's not worth it if there's a chance to trigger the cutscene.
The double rewind bug happens when you use a rewind the instant the prince touches the ground....at least i think this is the cause. You're probably getting it on the 3rd rewind, correct? That's the one where the prince hits the ground. My suggestion to prevent this is to wait just a little bit longer after the prince hits the ground (so use the rewind about half a second later than you're currently using it). Practice this and see if you can still pull off the trick consistently, and also if it prevents the double rewind bug.