If keeping your fort alive for some time is the priority, just use some serrated disk traps (with a chained puppy to make it work on ambushes) until you have the time and dwarfpower to build something proper.
If you want to actually build this trap before the first goblins arrive... well, make it your second priority right after producing food and booze. When everybody does some mining, masonry and mechanics, it definitely is possible to do it in your first year.
About the miners dropping eachother down the pits: There are safe ways to dig. You could designate each layer for channeling, starting at the top and only start the next when the previous is completely finished. Or designate the whole thing as up/down-stairs with up-stairs at the very bottom, and let them dig it out. Afterwards you need to designate the stairs channeled, again starting at the top and not designating a new layer until the previous one is finished. Water can flow through the stairs with no problems, but enemies may not be flushed properly through them and just drown. A plain old drowning chamber may be more effective and gives you the chance to collect their equipment once the water is drained.
Draining into the caverns is one possibility, but you could also drain it off the map if you're near the edge. I like to carve a large artificial cavern to drain the water into. If it's big enough, the water will spread out and evaporate, leaving mud for an underground tree farm.