When do I have to transfer the first month’s rent and the security deposit?

Both the deposit and the first month's rent must be paid into the landlord's bank account before you move into the apartment. Please find more information here.

You can find the details of the rent payment in your rental agreement. 

Wunderflats is not a contractual party to the rental agreements. Therefore, the rent or deposit can neither be transferred to Wunderflats nor refunded by Wunderflats to the tenants.