Does Preview remove the password permanently?

The exported copy has no password. Keep it somewhere safe, and keep the original if you still need the protected version.

Is there a way without Preview?

Any Mac PDF app that can “print to PDF” achieves the same result once the file is open.

Only try to open files you’re authorized to access. If it’s yours, guided recovery from hints is the route.

macOS has everything you need to open and even remove a PDF password built right in — no extra software — as long as you know the password.

Here’s the quickest route, and your options if you’ve forgotten it.