Gmail – Check Space Used by Label or Folder

If the skip inbox is checked, It should work ... it is working fine for me.

Maybe you should check the inbox setting

Is your priority inbox is enabled? I know that if it is enabled, it can show some mail not in the inbox as important.

You want to use labels for this. Gmail doesn't have folders, it has labels.

You can automatically assign a label to an incoming email via filters. The filter can even archive the email, so that it retains it, but doesn't include the inbox label. Archiving is just the removal of the inbox label.

You can also have gmail retrieve email from a third party account. That function can also attach a label and auto archive.

A cool feature of labels is that a message can have multiple labels, thus allowing you to avoid the problem of trying to store a message in two folders. You can also have the list of labels in the sidebar, making it easy to jump to a label.