Add the mailbox to the Gmail app
- Open the Gmail app, tap your profile picture and choose Add another account, then Other.
- Enter your full mailbox address and choose Personal (IMAP).
- Enter the mailbox password.
- For the incoming server, enter the IMAP server name, port and security type shown for your mailbox; the username is the full address.
- For the outgoing server, enter the SMTP server name, port and security type, require sign-in, and use the same username and password.
- Finish, then send yourself a message to check both directions.
Your server settings are shown in three places: on the page right after the mailbox is set up, under My mailboxes in My account, and in webmail under Settings → Apps & sync. Other Android mail apps ask for the same values: see IMAP, SMTP, CalDAV and CardDAV settings.
This adds your mailbox to the Gmail app; it does not copy anything into a Gmail account. To bring mail over from a Gmail account, see Move your mail in, and take it out. With two-step verification on, use an app password.
Calendar and contacts on Android
Android syncs CalDAV calendars and CardDAV address books through the DAVx⁵ app. In webmail, Settings → Apps & sync shows the CalDAV and CardDAV addresses with a copy button and the steps for Android: add an account in DAVx⁵ with that address as the base URL, your full mailbox address as the username and the mailbox password (or an app password).