Whatever you gave as emergency contact info is still on file, as long as that file is maintained. How long that is depends on whether it's a private provider or via an agency. Since you are not in that provider's care, it's unlikely they would ever know if you're experiencing an emergency. An emergency generally means you're unable to make decisions regarding your own care. For example, if you broke a leg and were conscious, a hospital would set it and send you on your way; they wouldn't call anyone unless you specifically asked them to. That assumes you're over 18 in the US.

As a general rule, therapists are adverse to appearing in court and do so only when subpoenaed. The exception to that is in the case of juvenile and domestic court, and even then it's rare and only happens upon request, doesn't even mean they actually testify. Another exception would be if a therapist's client asked them to appear on their behalf.