With regards to my current setting, Riftsea (I'm not playing it with D&D, but I aim to have the option available when it's done):
An eclipse?
If it's scheduled, that's all good (eclipses happen every 24 hours, for 12 hours, except for every third such period - it's how night happens). If it's been predicted, and there are newssheets in the region, people might go 'ooh, ah'; if it's a region without readily available public astronomy, there will be much religious kerfuffle. If it hasn't been predicted, everyone will panic.
Earthquake? Volcano?
Skyrealms are used to subsidance (there's not much holding up their continents), so the primitives who live there are relatively fine with them. The civilisations on the surface have never met earthquakes, as the ground is almost perfectly stable. Volcanoes are only encountered as massive portals to the Cardinal Fire plane, and thus probably enacted by massively militant mages who have made their presence felt earlier. This has the unusual side effect of primitives knowing how to deal with earthquake safety.
Bizzare weather?
Due to preponderancy of strange races and relatively public powers (mages, technomancers, telepaths etc), even little spiders falling out of the air isn't utterly unthinkable... just very strange, and people will want answers from their local power users. Storms aren't unknown, and skyships will seek safe haven at such times. Forecasting isn't as good, but communications can help people get to harbour before a storm catches them - if it's a natural storm.
A comet?
Strange things in the sky are a lot more common around here (angels, demons, dragonriders, skyships of three or four flags etc). This one is just high and uninterested in the ground. Of course, if it were a real comet, the shape of the world becomes important - it would actually approach from underground and cause a small earthquake, and then continue existance as a cloud of dust in the ether, with nobody ever seeing it.
Has your world become blase towards natural disasters because the inhabitants can cause them or are these understood happenings? Do they freak out because the gods cause them? (Do they?)
Those disasters that mortals can cause are regarded about the way we regard nukes today. Not exactly blase, but it's a different attitude - if something happens, the diplomats had better warn us first. Gods, well, it depends on the gods you follow - the Pales of Cathedra are the most able to cause disasters, but they're very close to the world and if they did, their followers would all get very depressed and anxious to please them anew. The multitudinous Angel Queens of the Ark would get an 'OK, we'd better deal' from their followers (and, in fact, have been known to blow up whole districts when their bioengineering projects got out of hand). Other deific figures are either dead or philisophical or asleep.
Ruling out the obvious (fixing things in an earthquake, avoiding a flood, watching an eclipse) that modern people do, how does your world or the members of it react differently?
Most people are blase about eclipses, as they're the basis of night. People from eternal shadow or eternal day would get very upset at first, but eventually learn to live with it. The response to damaging effects is largely the same, although most regions have magewrights to help fix things, so construction goes faster. In Brightsea, the cities stand atop huge pillars so they stand above the waves, so earthquakes might wipe out huge chunks of their nation. In the Ark, everything's built from living material or primitive clay, so Remakers would just engineer a new house and grow it to maturity once the situation was over. Actually stopping a disaster, natural or political, is really difficult unless you're on a political scale yourself...
