It sounds to me like a rule/filter is acting on those messages, so that would either be on the Gmail side or via another service like Sanebox if that’s active. Mail on iOS doesn’t have any sort of rule capabilities that I’m aware of, so something must be happening further up the chain.
As an iPad user only I don’t use the web interface. I strictly rely on the mail.app on iPad and iPhone.
I know that mail.app and gmail has a rocky relationship. I have not set up any rules in Gmail. It’s a completely new email account. So I think the fault lies with how Mail.app handles Gmails folders.
Therefor I’m thinking someone else most have experienced this issue of emails weirdly showing up as unread in the Archive folder.
@marius idea is worth checking out. Since you are iPad only, use the Gmail app to troubleshoot. I keep both it and mail.app set up on my iPad & iPhone (because gmail.app’s search is much better than mail.app).
I have several gmail accounts running, and have not seen this issue yet.
What I would suggest is removing the account, and re-adding it.
Do it under a different name so it does not restore the previous cache!