Gmail with Apple Mail: Remove Important and Archive labels?

Hello hmac11,

Welcome to Apple Support Communities. For information about "Important" folder on your Gmail email account, please check this out:

Hide importance markers in Gmail

1-Using a browser, open Gmail. You can't change this setting from the Gmail app, but the settings you choose on your computer will apply to your app too.

2-In the top right, click Settings   See all settings.

3-Click the Inbox tab.

4-In the "Importance markers" section, select No markers.

5-At the bottom of the page, click Save Changes.

Importance markers in Gmail

Regarding the "Achieve" folder, please check out the "Apple recommended" section in the following post:

how do i remove archive folder in mail?

In the meantime, you can try to move the messages in your "Archived" mail box, then delete it by following these steps:

Move archived messages back to your inbox

1-On your computer, go to Gmail.

2-Find the archived message.

3-Next to the message, check the box.

4-At the top, click Move to Inbox .

Archive or mute Gmail messages

Delete mailboxes

WARNING: Deleting a mailbox permanently deletes the mailbox and its contents (including other mailboxes), and can’t be undone.

1-In the Mail app  on your Mac, select a mailbox in the Mail sidebar.

2-Choose Mailbox > Delete Mailbox.

If you have problems deleting a mailbox on the server for an IMAP account, contact the provider of your account.

When you delete a Smart Mailbox, the messages it displayed remain in their original locations.

If you use the Finder to delete a mailbox, the change may not appear in Mail.

Note: If you set up Time Machine to back up files on your Mac, you can use the backups to restore content that Mail downloaded (based on options set in Mail and on your Mac). See Restore items backed up with Time Machine.

Create or delete mailboxes in Mail on Mac - Apple Support

We hope this information is useful. Kind regards.