Should the file permissions that Coppermine installs be set to 755 or 777? Is it safe to set them all globally like that or should certain files have file permissions other than 755 or 777?
755 is more secure than 777 (generally speaking), but usually you have no choice - chose 755 if it works for you on your server, if it doesn't, choose 777.
The script needs write access in some places, that's why you have to CHMOD. If you're paranoid, you could CHMOD the include folder back to 644 once your install went successfully and the config file has been created.
GauGau