Reversed. Yes. The input to the gate has to be the voltage across the capacitor to ground since that's the logic reference.
If you need a NAND somewhere else, then use a NAND. Just tie two inputs together and you get an inverter.
If you need an EDGE, you won't get one without the schmidt trigger. I had to change a chip on a single board computer to a schmidt because of reset issues and a linear regulator. If I used a switching regulator it would reset every time. With the linear, it wouldn't.