Deleting old mails

Is it possible to set up a rule which deletes, either on user action or, because of a ‘days since received’ setting?

This probably doesn’t exactly answer your question, but you should look at the possibilities of automatic archiving.
Take a look here: https://www.emclient.com/webdocumentation/en/8.2/eMClient/Default.htm#E-mail/Archiving.htm?Highlight=local