Spam is spam whether it is used by others for nefarious means.
Exactely what i have been trying to say, any unwanted mail is spam And there is usually only one button to use if you want to report it, SPAM or Unwanted mail. Anyway case closed on this
Once her emails are downloaded they are not synced anymore. [b]Especially since she has Chrome always opened to her gmail [/b]account online [b]and does not download her email using any email program[/b].
if you look at the info you posted from the avast message, there is a line that say

Process: /Application/Mail.app/Contents/MacOS/Mail

Did she previous use the mail app?
Is it still innstalled? … try uninstall