When I send an email to a group or an automated appointment reminder is sent to a group of clients for a group appointment, will each person's name that is in that group be listed on the "To" line ...

No. Let’s say there are 10 people in your group. When you send an email message to that group, only the recipient will see their name displayed in the "To" field. The system does not "CC" the other people in the group on the same message. It is actually as if the system is sending 10 separate messages. So no one in the group will know the other people in the group by way of names being included on the "To" line.