Is it possible to hide it when users look at their own profiles?

You should be able to just surround it in an IF substitution to exclude it from self. Example as follows.

[cb:if user_id!="myid"]BUTTON_HERE[/cb:if]

See the below tutorial for further usage information.

www.joomlapolis.com/documentation/279-community-builder/tutorials/18353-using-substitutions-throughout-cb

I click the button with the custom link and it takes me to a blank page (Header/footer only).

Which button? The one with the below link?

index.php?option=com_comprofiler&view=pluginclass&plugin=pms.mypmspro&action=message&func=new&to=[user_id]

There's no reason for it to error with "Message does not exist!" which such an error doesn't exist as it's "Message does not exist." (period ending not exclamation). This error is only thrown when trying to read or edit an existing message. There is also "Message not specified!" which is thrown when trying to save a PM with no message supplied.

Looks like it's an issue with the custom PMS routing when SEF is enabled. Was able to confirm and have added a bug ticket to fix.

forge.joomlapolis.com/issues/7399

Fixed and released as part of latest build. It was due to username not being SEF safe so it'd use only the user id, which it would think is a message id so it'd end up trying to read a message instead of create one.