the one that bugs me immediately is his cell battery is dead so he goes into a store, gets another battery and gets his phone working. I've never bought a replacement battery for a cell phone that didn't require you charge it before you can use it?
Dude, it's Manhattan. Haven't you heard? You can find a charged cell battery at 2 a.m. in the middle of a catastrophic monster attack.
Also, that monster was probably there for many months, walking down Broadway completely naked, and no New Yorkers noticed because they're above that sort of thing.