Do trainers see all members?
No. Only the members of the groups they run, and only while those members are in the group.
Can a trainer see what the club earns?
Only if the club lets them. Payments can be hidden, limited to those the trainer recorded, or show that service's whole share.
Can trainers create sessions?
Yes, single or repeating, with the same dialog the club uses.
Can a trainer send invoices?
If the club allows it for that service: email an invoice, send a reminder, record a cash payment or issue a credit note. Invoice runs they prepare stay drafts until confirmed.
What if someone is both a trainer and a manager?
They use one login and pick the role when they sign in.