To add to the already good answers, I'll address each piece that you brought up from the books. This answer will be completely behind a spoiler tag because all of the questions and answers contain potential spoilers:
1. Fain is Darkfriend, with special powers given him by the Dark One at Shayol Ghul
A: He was given special powers, but those powers were specifically designed to allow him (and compel him) to track down Rand, and isn't related to the True Power, which was only given to Forsaken
2. Possessed by Mordeth, who's evil and powerful (how?) but opposed to the Dark One.
A: Mordeth's power came from his approach to fighting The Shadow, which was to use its own tactics and methods against it. This lead to growing fear and suspicion in Aridhol, which manifested in the dark force known as Mashadar. The power of Mashadar (and Mordeth by extension) was not rooted in the One Power, but in that corrupting power of fear and suspicion.
3. Do either of the 'powers' refer to include the ability to channel saidin? In book 2 he uses a very conventional hammer and nails to kill a Fade
A: He is able to kill a Fade in that way because they fear him as much as men fear Fades, presumably because of his connection with Mashadar, which kills both normal creatures and Shadowspawn alike. While it is possible that a channeler could use the One Power to hold a Fade in place in order to nail him to a wall, this doesn't seem to be the case in this situation from the description given
4. but later on (in Caemlyn, after hearing about the Grey man attack on Rand) he does something very strange to a child
A: I'm not sure what specifically you're referring to here, so I can't answer this one
5. In Far Madding creates an illusion of a scene from the past to distract Rand, which seems to be something you can only do by channeling. But we've never (AFAIK) actually seen him channel. Maybe what happened in Far Madding wasn't really created by him, but only one of those weird attacks Rand has from time to time?
A: Far Madding has a Ter'Angreal called The Guardian, which blocks access to the One Power, so this can't be a result of channeling. The Guardian doesn't block access to the True Power, but as I mentioned earlier, access to that is restricted to the Forsaken, and Fain is certainly not one of the Forsaken. It could be unrelated to Fain, but given the context, that seems unlikely.
6. And how did Mordeth manage to live so long in Shadar Logoth?
A: Mordeth lived in Shadar Logoth without aging because his existence was as a part of Mashadar, which is not a mortal entity. Once he bound himself to Fain, it would be reasonable to assume that them together might be mortal, but we don't find out since Fain ends up being killed towards the end of the series. The part of Mashadar that stayed in Shadar Logoth was destroyed when Rand was cleansing the Taint on Saidin, but that apparently didn't have a major effect on Ordeith (the name and personality taken on by the combination of Mordeth and Fain).
7. Immortality is something we've only seen in the Forsaken and possibly female channelers who haven't used the Oath Rod.
A: That isn't strictly accurate. Channelers (men and women) age more slowly depending on how strong they are in the One Power, and how often they access it, but none live forever. Wise One channelers do eventually die of old age despite not using the Oath Rod, and the same is true for Windfinders and Kinswomen. It is suggested that strong channelers can live for many centuries, but not indefinitely. The Oath Rod cuts this length approximately in half, though removing the Oaths seems to restore their remaining life to some unexplained extent.
Male channelers in the Third Age simply aren't able to take advantage of this slowed aging as they mostly become completely mad before it would have an effect on their lifespans. Those who go mad either kill themselves or are killed by the Red Ajah, and those who are captured and severed usually kill themselves as they lose their will to live.
The Forsaken are also not necessarily immortal. They are all powerful channelers, so their lifespans (if not killed outright) would presumably be very long. That said, the reason they are still alive after such a long time is that they were sealed away with the DO. Ishamael was occasionally able to reach the world despite being sealed, but not enough to age significantly during that time period. We do see however that those who were closer to the surface of the seal (Balthamel and Aginor) were aged and degraded significantly over the many years of being sealed. That may or may not have been entirely due to aging, but clearly they were not completely immortal.
The DO offers the Forsaken immortality, but it isn't clear that he is actually able to follow through on that process, or if it was just an empty promise used to lure people to The Shadow. That immortality could also be in the form of the DO's reincarnation of the souls of Forsaken into new bodies. If so then he is technically telling the truth, but not in the way the Forsaken think he is.