HI and welcome,
The evaluation process can vary from several visits to just a few, depending on who does it and how it is done. Ours was several visits, bloodwork, EKG's , various testing ,allergy testing
A child may HAVE ADHD, but it is not what they ARE. Never tell a child they ARE ADHD.
My best advice for staying calm is to leave the room when things are at their worst. Sometimes I just have to get away or I'll lose it and yell, which doesn't do any good for anyone. Walk to the next room, get focused and then return when you feel you can talk calmly.
As far as the process, it really can vary a lot. Our pediatrician sent us to a child psychologist first because our son was only three when the ADHD was becoming very apparent. We got some behavioral tips from the psychologist that didn't work, so we were sent to a behavioral therapist and got some more tips (some helpful, but nothing really solved the problems). When things just got worse, we ended up meeting with a child psychiatrist. It took quite a while to end up with the psychiatrist. She looked over the history, including feedback from us and our son's teachers, and got him on a treatment plan with medication. He was very young at the time but a danger to others, so we'd exhausted options at that point.
Now, we meet with the psychiatrist regularly when medication stops working or just to check in. We also remain in contact with the behavioral therapist when issues come up.
You're very, very lucky to get in so quickly with a specialist! Good luck with everything!