) How can I find a cave spider, fast? I have no silk and he demands it I don't think I've ever seen a cave spider.
You'll rarely actually see them, because they're vermin. Mostly you'll see the webs they left behind. Try the caverns (all 3 layers). If you see webs in one of them, you can issue a "collect webs" job from a loom (the game automatically issues these if you didn't turn that option off).
You may need to send the military to explore the cavern and reveal all of it, lest you get spammed by pathing failures for webs that the weavers want to collect (they have X-ray web vision) but which aren't pathable (they can only see un-revealed webs, not un-revealed cavern tiles).
In my experience the best way to find cave spider silk for early moods is to have a) a really good bookkeeper, and b) have some miners handy.
Step one is the part that requires the book-keeper. On the stocks menu Spider Webs are listed, with the thread. If your book-keeper has done his job you can hit tab, to switch from listing quantities to listing individual items. Then you select a web and hit z. Now you know where the Cave Spiders in your map live.
It is very very important you picked a cave spider and not a Giant Cave Spider, because cave spider are vermin, which means they teleport anywhere in their home biome. Now all you have to do is dig a room somewhere in that biome, Cave spiders will magically teleport to that room, and leave their webs. Add a spare loom, and a guy with web-gathering enabled, and you should have at least one silk thread gathered in a few weeks, which you can spin into silk cloth if that's what your moody dwarf needs.
There are a couple caveats. You may need to have opened up the caverns before the webs show up on the screen. And you have to be quite careful in how you dig your room, because if you create a way into your fortress you'll be constantly invaded by cavern-dwellers.
Nick