I don't know if it really saves time or anything, but I always hug to the right during the entire level and everytime I do, I never seem to get a second "save me" section. Instead, I get the "Thanks fox!" animation from my companions.

Is this anything that could help/hurt the run?

That's actually what I've been getting lately as well. In my last PB, I hang to the right, but come back once the first "save me" message comes up. I then hug right again until I get to the point where I can grab the Twin Blasters and then I hug right again. That's when that second "save me" section turns into "Thank you!" I've definitely noticed a time save there with that method.

Yep, I do the exact same strat. Sometimes grabbing the checkpoint for health/marathon safety.