Yeah, it won't work. It works in every other developed nation in the world, and they all pay less money for better healthcare outcomes, but hey, the US is Totally Special.
The cost side works in every other developed nation? Surely you're aware of how much more expensive health care costs are in the US, so what might you be referring to?
There are other effects as well - nobody is getting away with billing $600 for a 5 minute visit, for instance.
Which isn't to say that single payer wouldn't be an improvement on the current system. Just don't get your hopes up for how much money it'll save.
No, we don't. Medicare is not single payer, though it has (for a subset of the services covered by Medicare) a default public option; it also has private, partially-public-subsidized pland; Medicaid, at least in many states, is not single-payer, either, even at the state level, even before considering overlap with Medicare and other insurance.
Apparently, states are unable to restrict entry into state benefit programs such as healthcare. They can't set up a waiting period for people moving into the state before they are eligible; that was declared unconstitutional by a previous Supreme Court. So if states were to do their own single payer system, in theory sick people could move into the state and immediately gain free healthcare simply by virtue of being a resident. Conversely, they could then move back to their home state that doesn't charge higher taxes.
So it wouldn't be feasible on a state-by-state basis unless a future SCOTUS reverses their precedent. This is unlikely in the short term, given the current Court's make up.