Do emails sent internally between users on the customer mailbox server appear in the Failsafe Mailboxes?

Not by default, but this feature can be configured. If the customer configures their internal mailbox servers to route all email for internal users based on DNS lookup which is not the default configuration for most mailbox servers (most use an internal directory and will route emails locally for efficiency reasons). For MS Exchange users, the Exchange Server Transport and Routing Guide (opens in new window) may prove useful.