R.G.
Yes, every daycare I have ever used, whether in-home or corporate, has required payment on the first day of the week for that following week. Most have also requested a deposit. This is not unusual at all. Now we pay monthly on the first of the month for the upcoming month. They do it so they are at least a little covered if you vacate and they can't fill the spot right away...
I would not be concerned about the director living across the street - she's always accessible that way, even if she's not on the premises...