No. Nope. Negative!

1. You can keep your email as long as you want. We even have a "closed" status you can select for mailboxes that are inactive (for example, for employees that have left), but whose data you'd like to keep at a reduced rated.
2. Free-of-charge, we backup all of our clients' data in our shared hosting environment so that clients can have data restored by request as far back as 30 days. Restoration is also free-of-charge, as long as it's not abused (it should not be a regular need). This means we can restore email, calendars, contacts and attachments, even when you've deleted them, assuming that we have at least one back-up of the information (for example, one night has passed between when you receive, or create the information, and when you delete it).

Archiving and Discovery is an extra, optional service, typically for public companies or those organizations that work for public companies subject to SOX compliance requirements. It not only archives data, it prevents users from deleting data, and makes it searchable across mail-boxes, by keyword, including exploded headers and attachments, for as far back as the archives are kept.  Not necessary or desirable for most companies, less than about 10% of organizations in our shared environment have enabled Archiving and Discovery.

Also see, What is Zimbra Archiving and Discovery (ZAD)?