Unfortunately, we just don't know the answer to any of those questions, though I think we can make an educated guess on at least one.
To start with, Strange's first battle with Dormammu in the comics did not involve the Eye of Agamotto. (I believe he was actually given the eye by the Ancient One because he defeated Dormammu). In the comics, when Strange goes to the Dark Dimension to confront Dormammu, Dormammu's retaliation ends up releasing a horde of Mindless Ones (very dangerous extra-dimensional beings). Dormammu needs Strange's help to stop them, and in exchange, Strange demands an oath to leave Earth alone.
In addition, the way the montage scene is written, its clear that we're not supposed to know exactly how many times it happened; I haven't seen any leaks of early scripts of the movie to know, but I'm skeptical that they ever gave a specific count. The best we can do is just count the number we specifically see on-screen, which is around a dozen or so.
However, it does look like both Strange and Dormammu do remember each loop. It's obvious that Dormammu remembers; the whole point is that, after enough repeats, Dormammu gets frustrated and makes a deal just to escape. If he didn't remember the previous iterations of the time loop, there's no reason for him to ever get frustrated. Also, note that Dormammu tries something to kill Strange different every time, implying that he knows his previous attempts didn't succeed.
But Strange, too, changes his behavior as his loop repeats. In his opening line to Dormammu, he starts to sound bored and frustrated, which only makes sense if he knows he's done this before. Also, IIRC, he seemed prepared for Dormammu's various attacks after they'd been used on him once, which only makes sense if he remembered them.