How does Smaug know about the Ring?

This is a very interesting question, however things are quite different in the book and in the movie, the short answer is that Smaug probably didn't know about the ring nor Sauron, the long answer is explained below.

In the book Bilbo entered in the mountain 3 times, the first time Smaug is asleep and he steals a golden cup, he then returns to the dwarfs who convince him to enter Erebor again. The second time Smaug is awake and angry for the theft, he talks with Bilbo (who tricks him into showing his weak point) and despite talking about the cup he never said the exact sentence you quoted from the movie. The third time Bilbo enters the mountain together with the dwarves.
So I'd say that in the movie Smaug is referring to the ring, even though it isn't explained how he knew about it and this precise scene didn't happen in the book.

The second question is even harder to answer because, even in the extremely detailed universe created by Tolkien, very little is said about dragons (only 4 of them are actually named).
What we know is that dragons were originally bred by Morgoth and that Sauron (who was Morgoth's disciple) used them in the famous war of the rings so some dragons surely knew Sauron and the rings (4 of the rings given to the dwarves were melted by dragons).
We don't know a lot about Smaug, he is the last great dragon on Middle-Earth but there is no way to tell for sure whether he knew Sauron and the rings.

I'm not sure whether Smaug says something about the darkness incoming in the book (he doesn't in the dialogue with Bilbo, but I'd need to check whether he speaks in other parts of the book too), but he can probably just perceive it, since he can feel magic and Sauron is gathering a huge power.
I think this (and maybe Smaug knowing about the ring too) are details added by Jackson to contextualize the tale told in "The Hobbit" inside Tolkien's universe and timeline.

I confronted the most important details with online sources, but it has been a long time since I read "The Hobbit" and "The Silmarillion" so I may be missing some (potentially important) details

What Smaug says in the movie:

"Come now, don't be shy. Step into the light. Hmmm, there is something about you. Something you carry. Something made of gold, but far more preciousssss."

In the book Smaug says nothing of the kind. Instead, he tries to get information out of Bilbo and sow him with suspicions about the honesty and intentions of the dwarves (the dwarves said nothing about how he would get his "14th share" of the gold back to Bag End, for example).

The movie script is shorter, more dramatic and is meant to foreshadow the part the ring would play in the subsequent adventures.

If there is a rationale for how Smaug detected the ring, it would be that as an ancient and evil creature he could sense its power. In the movie a parallel is drawn between the eye of Sauron and the eye of Smaug, the two images being juxtaposed. In fact, it is the gaze of Smaug's eye that pressures Bilbo and leads him to take off the ring, with the result that Smaug can see him. Presumably Fran Walsh, the screen writer, had the idea that, like the minions of Sauron, Smaug would sense the ring and be "drawn to it", almost like a ring wraith, or at least be aware of its nature at some level.

As for Smaug's knowledge of Sauron, this is not mentioned in the book, but of course a dragon would know such things. Smaug's abode in the Iron Hills adjoins Mirkwood and he would know of Mirkwood's descent into darkness, and the source of the evil power residing in Dol Guldur in the south of Mirkwood.