>The DNS name that comes up for the virtual IP address in the dashboard?
Correct. Otherwise the automatic certificate name won't match.
>Should there be [near] zero downtime with anything on the one-to-one or one-to-many NAT rules if there is a failover event?
No state is synced between the two MX. When a failover happens, all connections are dropped and must be made again. In the case of AnyConnect, users need to reconnect.