There's no telling how long that will last. The Black Tower was, in the same Foretelling, to be rent in blood and fire, and sisters walked its grounds. Does that mean they are going to be there forever, or that the Asha'man will never unite? The first part of the Foretelling might only mean their condition right before Tarmon Gaidon, and "stronger" could refer to many things beyond their influence abroad, it could mean simple numbers or abilities, even while their political power wanes.
No, they promise that Rand will give them as much. In the years to come, steamships will take it back. Again, not an eternal condition.
The books make clear that the merchants have access to noble-like wealth. There are also questions regarding the Tower's finances, with a much larger population to support, and fewer resources available to sponge, as they did with the Borderlanders. With other options for supernatural powers, people are going to be less inclined to hand over money. Also, their city's trade has been seriously harmed. Nothing in the books about any sort of solution to the cuendillar chains blocking the harbors...

Drought and heat are not exactly problems for people who live entirely at sea. Storms might be bad, but they are not the killer that drought is for an agrarian-based economy.

Not when the chance is being offered by representatives of a group you suspect of active hostility, and know to be pathologically dishonest, self-serving and incapable of dealing in good faith. Imagine you're a Jew hiding in Nazi Germany, and a bunch of Nazi officials casually show up at your hiding place, telling you they want your kids to assist their scientists with a solution to global warming, and BTW, they know all about your secret underground network that you have worked so hard to conceal all these years, for your peoples' safety. Nynaeve and Elayne are somewhat lucky to have left the boat in one piece.

Even worse. Even Egwene admits that failure to accept responsibility or meet debts for wrongdoing exacerbates the evil. Someone who wants to coast along without paying anything is not any better.