At this point, it's the former. If there is any similarity to his ability to sense women channeling, it's probably psychosomatic. At this point he can't feel his own type of Power yet, let alone the opposite.

Rand is probably not used to the concept of fame. He has spent all his life in a village where everyone is either known or a stranger. These are the only two categories in his experience. Lan might not know Logain personally, but he is probably familiar with him or knows his reputation. There has been no further evidence indicating actual association of the two.

Rats are not domestic animals, and as such have reflections in T'A'R. You suffer the same injuries in the real world that you do in the dream. He broke the backs of the rats in the T'A'R inn. An alternative is that he had Rand in a personally created dream, and subsequently killed the rats in the inn to remind Rand and freak him out. Recall that at this time he is trying to intimidate Rand into following him.

Egwene notes, in LoC, I believe that you can bring a dreamer into a private dream, which the Wise Ones decry as evil and a tool of the Shadow. It stands to reason that this is what Ishamael is doing. In these dreams, it is the Dreamwalker who has the power, rather than the ordinary sleeper (as in Gawyn's dream), hence the attraction of this technique for his purposes. He could have brought Rand to his own created dream or to a T'A'R reflection of a place that truly exists, either is plausible.

Possible reference to his association with Mat & the latter's Illuminator projects, or getting the credit for the Mat's gunpowder weapons. Remember his "joke" to Elayne on the Sea Folk ship? That the next age might remember Thom as a powerful figure ("not juggling fire but throwing it about" ), is a wink of RJ's suggesting the origin of the Merlin legend as a counterpoint to Rand's obvious King Arthur. The Seanchan already mistook Thom for the man behind the Ebou Dar escape and kidnapping of Tuon, as well as the subsequent military victories. The use of gunpowder weapons by that same army could end up being attributed to Thom as a result, instead of Mat. Min's viewings, according to RJ, ALWAYS refer to the future. Thom's subsequent involvement in Tower affairs might be an explanation for that part of the viewing, as with his eventual marriage to an Aes Sedai.

The trees thing is as yet unfulfilled, but the broken crown refers to Faile and her relationship to the queen of Saldaea. In the prologue to PoD, it is stated that if Tenobia dies without issue, Davram Bashere or his heir will succeed her. Faile is his heir, and Bashere himself refers to the Broken Crown when he accuses Perrin of marrying Faile for her titles. The implication is that there was some controversy over who should have inherited the crown of Saldaea, perhaps with Bashere refusing to concentrate on his military duties or some other outcome, and that his family has as strong a claim as the royal family. It seems significant for Perrin as his marriage to Faile has him three heartbeats away from that same crown.

The face could refer to his nature, as that is the notable feature his image possesses in Ba'alzamon's presentation at the darkfriend social in tGH. The Red Eagle was the banner of Manetheren, and Mat's old blood will later emerge, marking him as a descendant of that nation. The eye on the balance scale is referenced in a dream of him ripping out his own eye to place on a balance scale, and the prophecy of the Finns that he is fated to "give up half the light of the world to save the world."

The Prophecy of Rhuidean foretold that Rand would pour out the blood of the Aiel like water on sand, and that same metaphor was used to describe one of his early workings of the Power in tGH. The black rock is as you say, and the white hot iron and begger's staff have yet to be resolved.