When started, should the car stay idling without the throttle cable attached?
 

Not sure about it

Not sure why anyone would want/need to do this, but the short answer is yes. Assuming the TB is properly adjusted per the FSM to begin with, and the cable is too, the cable should have a tiny bit of slack in it when at idle (foot off the pedal).

Well then someone before you adjusted the cable to put tension on the throttle body to hold an idle which is a big no no.

I'm not understanding the thought process here...this is a 2 minute experiment...why did you not reconnect the cable to see if your car would magically stay running?

Is that the way it should be adjusted, no, but it would confirm that's why the car is now dying.

Any Ideas?

Yes, have you read any of the stickies that link to the FSM (factory service manual)? There's also several rotary sites, including this one that document setting your idle and checking your TPS (throttle position sensor). I recommend reading up and getting those two items addressed, they'll likely clean things up.

Without a tb cable yes it should idle on its own. Does your car still have a functional BACV and Cold idle thermowax?

Check for vacuum leaks.

Put all the stuff back on there that's supposed to make it idle.