You can day trade online stocks in both the NASDAQ and NYSE. Apart from seeing the Exchange displayed in the various broker screens it's really transparent i.e. same process, same screens. So I can trade EW - Edward Lifesciences Corp [NYSE] in exactly the same way as trading MSFT - Microsoft Corp [NASDAQ].
Not exactly, no. Day traders operate in the (very) short term, but they're not restricted to any particular market or exchange (in the case of stocks). Their trades last from a few seconds to a few hours but, come bedtime, they're 'flat', meaning that they have no open or pending positions and, therefore, no exposure to overnight risk.
Swing traders carry trades overnight and, sometimes, over weeks. Position traders may hold trades for many months. These traders can wake up in the morning to find that their traded index, stock, commodity or forex pair has moved massively overnight and their day has got off either to a fantastic start or, possibly, to a disastrous one.