1)
A U.S. equities exchange has to be registered with and approved by the SEC. There are currently 12 of them. As a registered exchange you have to play by certain rules, and in exchange your displayed top of of book is subject to the order protection rule, meaning that no other venues (exchanges, dark pools, ECNs, or broker dealers) are allowed to trade through your top of book quote without first routing an order to you for the full amount of the displayed size. Exchanges also get a share of SIP revenue (which is important to exchange economics).
In it's most technical definition, a dark pool is an ATS (specifically not an exchange) that has no displayed quote i.e. no market data whatsoever. Most ATSs in the US fit this definition, though there are also ATSs that display a quote. For example, IEX is an ATS that has feed for their top of book quote... so technically not a "dark" pool.
NYSE doesn't run a dark pool and it would be weird if they did. Most US equity exchanges support fully hidden (completely non-displayed) orders, so if you want to trade in the "dark" on NYSE you can just send a hidden order. I think this is one area where futures exchanges and equity exchanges differ. In futures you have icebergs but not fully hidden.
2)
Wholesalers aren't allowed to give you a worse fill than the NBBO. That's the law. Often they will actually give you a tiny bit better than the NBBO on marketable orders. I think they do this so retail brokers can advertise price improvement. For most of the low cost retail platforms there is no ability to direct your order. You give it to the broker and I think basically if it's a marketable order they send it to the wholesaler and if not they post it to an exchange that offers a liquidity adding rebate (the brokerage keeps the rebate, natch). If you are willing to spend a little more on your brokerage account, you can use a platform like Interactive Brokers which does let you route orders directly.