This has nothing to do with standardization, it is about whether certain quantities are normally distributed. The assumption can be stated more simply as: the errors are normally distributed. Also note that there is not a separate set of residuals for each variable, as you suggested, rather there is only one set of residuals (for the outcome variable).

For a more detailed refutation of the quote you posted, see here:

http://www.talkstats.com/showthread....-TalkStatters!

In theory yes, but I have no idea how PROCESS is written or what exactly it does under the hood.