Ideling but not running

Mine does this when it's cold. It will idle, but if you give it any throttle it will die. I let it idle for a minute, kinda playing with the throttle, and after that I take the choke off and it's generally good to go.

Also, you may has a standard NT carb. Id there just 1 screw? That's the idle screw.

i would try a carb without the cut off on it and see wut it does. just put a new 1 on mine after i experimented with the 1 it came with and..well lets just say i played with too much lol. got 1 off ebay for $10. runs better than new now

I can tell by the muffler that it is a 2099 or earlier SKyhawk so has an NT carb.
The 2010's have CNS carbs with the exact same problem, but the NT's are a different cause.

NT carbs are simple, they don't have a separate idle circuit. All it has is the idle screw, float level, and needle 'pac man' setting itself.
I like that term for the needle clip by the way ;-}

An easy first test is turn off the carbs gas valve and CAREFULLY drop the float bowl to check fuel level in it.
If you don't spill some chances are the the float tab that moves the fuel flow needle needs a TINY tweak.

Second easy is pull off the air cleaner and exhaust pipe and try it with the Pac Man clip in the middle.

Though it will be loud as heck, if it goes then it's a clogged pipe or air cleaner.
If not, try another spark plug.
You said you soldered the CDI wires?
Check the plug wire itself, and verify you capped off the white wire so it is not shorting when it vibrates.

Basically narrow the problem by finding out what it's not, then put those parts back until you know for sure it is just the carb.

can you take a pic of the bike? I want to see what kind of angle the motor is on.

Ya had a ho-made read valve there! [snork] Glad you found it before the motor sucked it up. Silly stuff happens to the best of us lol.