What's the trick behind carrying the prince throughout the Crown Prince's Castle level? I've been mashing grab, but I keep dropping him at doorways and stairs, wondering if this is just exclusively an emulator problem.

Only trick I know is to mash faster :/ It's hard to not drop him no matter what you do, this is one of those cases where you need a little luck to not lose time.

Not making sharp turns seems to help, and you can also hug the wall in some places which makes it more likely you will pick him up again if you drop him, but neither of these really help in the doorways.