[Pam Korda, Leigh Butler, Timothy Itnyre, Jennifer Liang]
In [LOC: Glossary, entry "Moiraine", 710], it says "She vanished into a ter'angreal in Cairhien while battling Lanfear, apparently killing both herself and the Forsaken." That "apparently" definitely left the question open.
WH tells us that Lanfear was "held" by the Snakes and the Foxes, either before or during her transformation into Cyndane. Though becoming Cyndane could have involved Lanfear's death at some later point (see section 1.2.4), the "held" statement indicates that Lanfear could not have died immediately after falling through the twisted door with Moiraine in TFOH. In other words, the act of falling through the door was not an automatically fatal experience (though it doesn't preclude the possibility that Moiraine died later).
One piece of evidence which points to her possible death is the breaking of her bond with Lan. However, we know from [TPOD: Prologue, Deceptive Appearances, 28] that "being stilled snapped [an AS's bond to her Warder] as surely as death. One of Irgain's two apparently had fallen over dead from the shock, and the other had died trying to kill thousands of Aiel without making any effort to escape." (Irgain is one of the AS stilled when Rand escaped from the box in LOC; Flinn Healed her of stilling in WH.) This quote shows that stilling an AS has the same effect on her Warder as her death. Thus, Lan's reaction only indicates that his bond was "snapped," by death, stilling, or something else.
Besides stilling, the breaking of Lan's bond could conceivably have been caused by the shutting off of the Red Door into Finnland. When Moiraine chastises Rand and Mat for using the Tear doorway in [TSR: 15, Into the Doorway, 178-9], she says, "One of you would have been bad enough, but two ta'veren at once - you might have torn the connection entirely and been trapped there." If one substitutes "channeling combatants" for "ta'veren," one has a description of what happened when Moiraine and Lanfear went through the door. The "tearing" of the connection between the two universes may have torn the connection between Moiraine and Lan, as well.
Of course, the above is either circumstantial evidence of or conjecture about Moiraine's survival. The real evidence that convinces us Moiraine is alive is the various visions that imply that she will return.
These are: Egwene's vision [TFOH: 15, What Can Be Learned in Dreams, 214] of Thom pulling Moiraine's blue jewel out of a fire, and Min's comment in [ACOS: 35, Into the Woods, 543] that Rand would fail without "a woman who was dead and gone," which almost certainly refers to Moiraine, and her comment in [ACOS: 35, Into the Woods, 546] that "Moiraine was the only viewing of hers that had ever failed." She must have had one or more viewings about Moiraine that were not fulfilled, and since WE know that Min is never wrong, then Moiraine will almost certainly return at some point.
There is further confirmation of her survivial when Mat reads her letter to Thom which states plainly "I am not dead." [KOD 10: A Village in Shiota]. Moiraine claims her foreknowledge comes from a secret source, most likely the rings in Rhuidean. Whatever Moiraine saw in those rings, it made her believe there was a chance she'd survive. Enough, anyways, that she'd write out a detailed list of instructions for how to retrieve her and make arrangements for it to be delivered to her rescuers. Mat and Thom certainly believe that she's alive, if their conversations on the matter are any indication. [TGS 27: the Tipsy Gelding]
Furthermore, she has a "small shred of hope" before attacking Lanfear, so there IS hope for her future. This "small shred" is probably a glimpse that Moiraine got of her future from a source other than the Rhuidean rings, either from the Red Door in Tear, or more likely, from a vision of Min's (probably the one that predicted Moiraine's marriage to Thom - see section 2.5.3).
Several people, though, have wondered why, if Moiraine had a prophecy or a Min vision indicating she will survive the battle with Lanfear, did she behave as if her death were a fait accompli, and make arrangements accordingly?
The answer is, she's human, and humans have doubt, especially when it comes to our own mortality.
The Rings of Rhuidean showed Moiraine a lot of stuff, and then nothing after the scene at the docks. All the WOs said that the rings show a person's entire life (or possible lives). In between going through the rings and the showdown at the docks, Moiraine had firsthand experience of the predictive power of the rings. So, she had no reason to believe that she wouldn't end when the events she saw in the rings did.
If Min did have a vision involving Moiraine and Thom, that probably would not have been enough to completely convince Moiraine. Min's talent isn't thoroughly understood, even by Min. She sees lots of things she can't interpret. Given the later, highly convincing, evidence from the Rhuidean Rings that she was going to die, it's reasonable to suppose that Moiraine wouldn't have thought that Min's vision, whatever it was, had not meant what Min thought it did, or that it was false. After all, it wasn't like Moiraine was able to check back with Min after her trip to Rhuidean. And note that not even Min herself is convinced that her visions are infallible.
So, she arranges her affairs in the expectation that she's going to croak. However, there is still something which makes her think that she just might somehow survive. It's not enough for her to tell anybody, but it's enough to give her a little bit of hope.
Which raises another question: If she didn't die, then why didn't the Rings show anything post-Foxland?
Possibly the rings don't work across dimensions. Remember, they work by showing many possible futures for the person who goes into them-- the possible results of future decisions, etc. If the Rings can't see what Moiraine does/is going to do while she's in Finnland, they can't extrapolate to any later point, after she returns. Another possibility is that the rings are somehow linked to the viewer's channeling ability, and thus couldn't see past a point where she is severed (assuming Moiraine was stilled when she fell through the doorway, of course).
In [WH: 14, What A Veil Hides, 324] we are introduced to Tuon's damane, one of whom is a former Aes Sedai, now called Mylen. She is described as tiny (shorter than Tuon) and pale in coloring, and "half-dead with shock and fear" when Tuon bought her.
An awful lot of people immediately leaped to the conclusion that Mylen was Moiraine, based on this description. This idea, though, is completely false.
From [WH: 19, Three Women, 403]:
Teslyn, to Mat: "'The others do be...changed.' Teslyn's mouth tightened. 'Guisin and Mylen - I did know her as Sheraine Caminelle, but she do answer only to Mylen, now - those two would betray us.'"
Teslyn knows who Mylen is (i.e. not Moiraine), and there's no reason in the world to suppose she's mistaken.