Since his meeting with Taim in Cairhien, until he settled down in Far Madding, aside from the side trip to pick up Nynaeve & make babies, Rand was deliberately letting himself be seen in various cities, while leaving subtle hints that he was going to Far Madding. Demandred's reaction is exactly what he was going for. He wanted everyone confused about where he was going or what he was up to, but was leaving trails for people who were actually following (i.e. physical evidence that would not be reported via rumors or spies, but might be spotted by pursuers) him. As he told Min in the prologue, he was afraid of interference when he tried Cleansing saidin, so he was laying a trap for anyone who was hunting him, like the renegade Asha'man, while moving about to confuse other interested parties, and thus keep them off his back while he prepared his ambush for the renegades and anyone else who was seriously going after him.

Stashed in the palace and warded, IIRC. I can't remember if Rand or another character said as much, or if that was an RJ revelation. I think the latter because I have a firm recollection of him saying that Moiraine's horse was in the stables of the palace.

Yes. See above.

Negative. He mentioned that in particular, because it is not a city in any meaningful sense of the word. He said that Rand has been to a bunch of cities, and Rhuidean, not because it's important to him, but because it is not a city in his mind. This is not to say that he is not there, of course, but this conversation is not a clue in that direction.