Sorry I didn't reply earlier.
1) My 2004 parts book says that all the 450's share the same CDI, so that's good.
2) It does sound from the dark smoke that it might be running rich, but your jets are stock, though your needle position is different. What colour is your spark plug after a full-throttle blast?
From my owners manual:
Main jet: 175
Pilot jet: 30
Needle clip groove: 3rd (from top) Just to clarify, is your clip in the top or bottom groove (pointy end of needle downwards!)
Check the float height: it should be 11mm. (I checked mine a couple days ago and it was 13mm, but my 2004 TE450 runs ok and is stock apart from a 35 pilot).
Back in the day the '04 was infamous for being hard to set up. The tweak then was to replace the stock needle (a 10E1-52) with a 10E1-50. I have one on order. Stock jetting (or one size up on the pilot) seemed to work for most people, from the research I've done. The fuel screw should be about 2 turns out.
The other recommendation I've seen is to set the accelerator pump preload to zero.
Circled in the picture is a little screw and locknut. It should be adjusted so that when the throttle is closed the black cam should just rest on the accelerator pump plunger (the vertical rod with the plastic cap on it), but not press it down (as stock it has about 4mm preload). You can twist the black cam clockwise to see the amount of preload. The adjuster is a pain to get to - it's easiest to take off the e-clip and remove the arm. My preload was set at 1mm, so you may need to experiment a bit to get a good response when you open the throttle. Don't play with the accelerator pump until you're happy with your jetting though.
While the carb's apart check the float needle and seat are in good condition and the o-ring on the seat is good. With the carb full of fuel twist the throttle and make sure the accelerator pump jet is squirting straight down the bore. Another part that can give trouble is the rubber inlet manifold - it can crack and perish, so have a good look at that too.
Finally, a link to the 2004 parts book (it's got a good exploded diagram of the carb):
http://cid-7f257a1c95a683f8.skydrive.live.com/self.aspx/Public/2776TC-TE-SMR 2004 Parts Catalog.pdf
Edit: Don't forget to check the valve clearances before attacking carb settings!