With those symptoms on an AT, I'd start with the neutral saftey switch. When it won't start, try shifting into another gear & back to park (or neutral) if it'll let you. If it won't let you, just jiggle the shift lever a bit & try again.
If that doesn't work, you'll need to find the switch (sorry, I keep my distance from auto trannys) & test (or adjust in some cases) it.

Dave