There are a lot of gotchas here, and it may take some digging to find out exactly what's going on.
To start, and to clarify, you're using the standard Mail app on iOS, right?
That isn't clear from your post, but is kind of important to get right, since the following may not apply to other mail apps.
First is the nature of the emails being 'forwarded to [you]'. How is this happening, specifically.
IF the the emails are going to some other user, and that user is forwarding the emails, putting your address in the To: field, then these should come in like any other email, typically with "Fwd: " prepended to the subject line.
On the other hand, if this is a shared email account (where multiple users/devices are logged in to the same account) then any changes made by one user (such as flagging the message, putting in a folder, etc.) would be reflected on all other devices.
So that's the first thing to clarify.
Secondly, assuming the first model, above, all incoming mail will go to the InBox by default. There are only two things that can (at least typically) override this. One is spam filtering (where Mail.app thinks the message is spam and moves it to the Junk folder), the other is Rules (where you define a set of criteria to check and the rule moves mail to a specific folder if that criteria is met).
Of these two, spam filtering seems unlikely, since the messages are going into a folder called 'import-2' rather than the normal 'Junk' (although this can be overridden).
Despite you saying you have checked 'every permission, setting, etc.), that doesn't actually tell me you checked the Rules. Did you check that there is/isn't a rule setup to filter these emails?
If you're on iOS, there is no mechanism to configure mail rules. You have to either use a desktop system, or the web interface ( https://icloud.com/mail/ )
