The only one that might be a problem is RNRC since it goes upside down. Had my bag an everything I rode.
Actually, because of the centrifugal force, your bag won't fall out. However, if you want to feel more secure, you could put your foot through one of the straps. Especially since I go solo, I like to do that to make sure I don't leave it behind.
I'm pretty sure you can bring your bag on every ride. At least when I was there in November/December I was able to to my bag on TOT and EE!! Enjoy the parks!!