F5 Big-IP HA configuration with SSL certificate

If you are referring to SSL certs you uploaded in support of a service, for SSL offloading as an example, then as long as your active and standby LTMs are in sync the cert will be available on both devices.

No. While building trust it's require device trust certificate to communicate each other on secure way(exchange information). But during failover SSL not require.please elobrate some more to understand your requirement.