The view them as tools and animals. Even the reluctance to use Healing isn't universal, it seems to be a function of what that cultural perception of damane means to an individual. Some people love getting "doggy kisses" from a pet while others find letting an animal lick your mouth to be beyond disgusting.

Anyway, the issues with misspeaking among the Seanchan are all about presuming to speak. If a noble or owner or otherwise superior person tells you to speak, you do. It's not necessarily either the same type of Foretelling as wetlanders are seen to do, since Lidya gave Tuon hers in response to a request to tell her fortune (which was the reason why she remained under the veil under leaving Mat, which had a lot of ramifications for her entire arc in the weeks after landing, and due to her marriage, presumably the rest of her life) she was sei mosiev because she acted against the damane who was merely obeying orders. That implies that lashing out against a damane in such a cases would not be normal practices. An awful lot of people seem to bring their own preconceptions of how slavery works to blame the Seanchan for things we absolutely never see them do, or take extreme interpretations of comments or incidents. For instance, I read someone getting all in a high dudgeon because of Tuon's comment to Mat about staying away from the damane so no one thinks he's a pervert who likes to sleep with them. This commenter took the comment to mean that this was a common practice of the Seanchan, when Tuon's comment is pretty much exactly what you would say to someone from another culture who doesn't see the harm in hanging around a playground watching the kids. If boinking damane is such a common thing, or even acceptable to the Seanchan, why would Tuon say "You would not want anyone to think you're perverted. The fact that she felt the need to explain that this fetish even existed means they don't see it as typical or normal, and Tuon's words have a stronger connotation of severe social consequences, rather than a winked-at vice, whose forbidden status is only honored in the breech.

Anway, if a damane can tell a fortune upon request, maybe they use some sort of aid or drill to put them into an appropriate state of receptivity. Wetlanders didn't think unweaving was practicable. There's no reason that Foretellings can't be managed as well. In the case of the unweaving, it wasn't even a One Power technique, merely mental discipline and practice. I'd bet Elayne's aptitude for reverse engineering was the biggest contributing factor to her doing much better than Aviendha on their respective first attempts.

The variety of manifestations suggests rather to me that it just might be trainable, since there are plainly different ways to do the thing. If the form of expression can vary, what else about it can?
I don't know. At their first meeting, Rand noted that her treatment of his hands was reminiscent of Nynaeve's, implying that she was using the power in some way. She was never called a wilder or anything, nor was any references made to that incident (though IMO, her Healing Nynaeve's black eye from Latrelle while she was fighting Moghedian in T'A'R for the first time, might have been a way of settling that issue for readers who had that same interpretation), so it's impossible to say what she actually did. Maybe there is some way for a channeler to augment natural processes without actually channeling, somehow sharing a fragment of whatever it is about her that extends their lifespan.
But why would she then be put off by something which was already an area of interest for her?
I suppose with him, that goes without saying.
Yes and no. My point was that no one would NOT care about such things in such a society. Maybe some AoL types like Graendal (which might also explain why they were not as good as the Sea Folk - once they had the weather under control, not to mention climate controlled buildings, it became something they ignored). Not every woman who cared about the weather would want to go to the Tower, but every woman who would want to go to the Tower, would be one of those people who care about the weather. Because she is a person living in a pre-industrial time period.

In that she, and anyone else, had no idea it was even possible. The means would be unclear, but the desire present. Like making someone obey your wishes, or hear by remote, which are the two most common wilder tricks, and just as conceptually impossible as changing the weather.
No, they are not that much more likely. You're still thinking like a modern person, for whom only the most extreme weather is an issue. An unseasonal frost, or a spring with a little too much rain or a summer without quite enough, could ruin a crop, and even if you aren't going to starve, might have to do without a particular food item one year. A lot of teenagers would definitely have motivations to change the weather when your whole social life depends on being able to go outside. Don't you think there were a whole bunch of girls with crushes or general romantic aspirations who were hoping with every fiber of their beings that the weather would warm up enough to celebrate Beltine that winter, immediately before EotW? Being able to walk around in public showing off that braid was the culmination of Egwene's lifelong ambition at that point in the story! A cold spring that refused to come, or a week of torrential rain that kept everyone inside would have been a disaster of equal magnitude to a teenager as a tornado.
That might be a cultural thing as much as anything else. Altarans in general seem to thrive on telling outsiders where to head in.
That's not the same as weather control, that actually seems to be one of the go-to manifestations of channeling, and one of the most-cited combat uses, aside from fire. No one is going to know who did it for the actual weather stuff that the Sea Folk use, like favorable precipitation and temperature conditions. Hell, they probably wouldn't even notice a series of nice days, since most people tend to see those as the default. The storm staying away until Pa manages to finish fixing the roof on the barn, or the early frosts holding off long enough to get those last crops in - that's what people expect and hope will happen, because it has been known to do so naturally.

But why would that same mindset not apply to weather working?
What on earth do you suppose women do? Watch soaps all day? Go shopping? Women also do farm work. They are maintaining the family gardens and milking cows and caring for chickens while the men are out guarding the sheep herds and working in the tabac fields. The female role on a farm is typically far more diverse and generalist, while men tend to specialize in the major heavy labor projects. That's why in rural restaurants, they call a breakfast meal that is heavy on carbs & protein a "farmer's breakfast". The men had to do the huge and vital heavy labor or the relatively dangerous stuff, while the women did everything else. To suggest that women or children were somehow isolated from such things as the effect of weather on crops is a huge stretch.
Not really. Men would be more likely to think of farming in terms of the back-breaking labor, and how to shorten or reduce that, I would think. Remember Rand's manifestation of channeling in Egwene's test? He blasted a stump that he was having difficulty removing with muscle power.
And he sucks at the weather, which is probably unrelated. The clear sky is often cited as evidence of the channeling behind lightning.
She's not violent, and IIRC, Fire is a component of lightning weaves, so it would be more likely to occur to men for that reason.
But until someone like that shows up, why would a woman using the Power without knowing what she is doing, limit her choice of abilities for fear of a sister showing up?
And to be fair, I hate bringing in "RJ said so" as the reason for how or why something works or happens (not citing him as an authority, mind you, just begging the question of functions and motivations within the story itself).

Or a perverse fascination with the same, like firefighters who become arsonists.
But that's centuries, we're talking about. I'd think it was the sort of thing that came later in her career, when her rep and status prevented some victims from complaining, and caused them to be less listened to when they did mention it. If she indulged early, it would probably have been with her first big Healing cases, especially because she seems to consider her pleasures her due and a fair price for her services. And I just realized I am getting way too into the mindset of a psychopath & serial offender....