Can it unlock a PDF without the password?
Only a file that opens without one and just restricts printing or copying: those restrictions come off. A file that asks for a password to open is encrypted with it, and without it there is nothing to decrypt with. The tool never guesses passwords.
Is it safe to type a bank statement’s password here?
The file and its password stay in your browser: nothing is uploaded, and neither is stored. Close the window and both are gone.
Will people still be able to print or copy it?
That is what the permissions decide, and Acrobat, Chrome and Preview honour them. They are not encryption, though: anyone who can open the file can find a program that ignores them. For a file nobody else should read, give it a password to open.
Which readers open the protected file?
Every current one: Acrobat Reader since version X (2010), Chrome, Edge, Firefox, Safari and Preview. The file is AES-256, the cipher the PDF 2.0 standard recommends; RC4 is not offered, since it is broken.