Are my old invoices and payments moved over too?

Not automatically. What the CSV import brings in is students and their families; the invoice history of previous years stays in your old records, and that is usually the practical choice — old bookkeeping does not need rebuilding for you to start invoicing next month.

What does need to move is the state of TODAY: a student with sessions left is given a credit pack the size of that balance, and money parents have already put down is recorded as a deposit into the family wallet. Old arrears you still intend to collect become manual invoices with their own due dates.