re :On the 3 gmail accounts the Archive options in greyed out.

I presume you have a gmail imap mail account. I presume you are talking about 'Archive options' button in this location: Account Settings > Copies & Folders

Gmail stores all incoming and outgoing mail in the 'All Mail' folder. You see emails in other 'folders' like 'Inbox', 'Sent Mail' etc because gmail applies a label to an email. It is a bit like a tag. This means any email stored in 'All Mail' folder can be seen like a virtual view in any number of other folders if the label for that folder is applied.

When you delete an email, but do NOT put it in the gmail 'Trash' also known as a 'Bin', it is not deleted, only the 'label' has been deleted. This is usually more of a problem for pop mail accounts. When this occurs, gmail automatically 'archives' the email. This means the email is now only in the 'All Mail' folder.

If you choose to 'Archive' a gmail email, this means all labels are removed and the email is now only stored in the 'All Mail' folder. Gmail does this by default.

This means, in Thunderbird, there are no 'Archive Options' on how emails are stored in the 'All Mail' folder.

Gmail also apply the label 'Sent Mail' to all sent emails, so by default those emails are put into the 'Sent Mail' folder, hence why you do not need to select 'Place a copy in'.

A good article to read which explains a lot about gmail: https://support.mozilla.org/en-US/kb/thunderbird-and-gmail#w_gmails-particularities

https://support.google.com/mail/answer/78892?hl=en

re : New email coming in is automatically move to the Archive folder.

Q: Have you created this 'Archive' folder because it is not one of gmails default folders? Q: Do you have any Message Filter that mentions 'Archive' ?

Menu icon > Activity Manager Q: In the Activity Manager, does it mention any moving of emails from 'Inbox' to 'Archive' ?

Q: Do you see new emails arriving in the Inbox and then see them being moved? OR: Q: Never see them in Inbox.