eM Client provides extensive advanced email search but as of this writing does support exclusion of mail that has a gMail user labels.
Background. gMail does not use folders to store mail, rather it uses tags or user labels. That is gMail folder “All Mail” contains a copy of all every email, and there is only one copy of this email. Other “folders” in gMail are simply views or lists of the emails in “All Mail” with a specific tag / user label.
If an email in “All Mail” is deleted the tags / user labels are also removed, and the email is no longer viewable in the “folder” to which it was tagged / labeled. The email is gone.
As of this writing this only way I know of to exclude gMail emails older than a specific date and do not have a Google User Label [are not in an eM Client gMail Folder] is to enter the following query [modified to fit your requirements].
before:20130101 has:nouserlabels !in:inbox -label:sent -label:drafts
Credit for this soution goes to:
As to the copy within the eM Client, the sync process with gMail deletes the emails deleted in gMail.