Fuel system On first start engine runs ok with slight vibration. After running for about five minutes the idle begins to drop engine starts running rough then dies. Sometimes if you increase gas engine smoothes out but opon returning to idle runs rough and dies. Often once idle has dropped it will not respond to increased throttle.

Bleeding the fuel system is typically only needed on high pressure systems like GDI and Diesel vehicles, and only to get them to start if they run dry once.
If your engine keeps running but the idle speed moves up and down there are many things that can cause that. More information is always helpful. Does it always do it whenever the car is running? Or does it only do it conditionally (IE Only when 1st started in the morning, only after warmed up or getting off the dreeway, only with AC on or at night...describe as much as you can.)
Most newer cars have computer controlled idle strategy that wants to idle the car cold around 1000-1500 rpm (After an initial higher rev called a FLARE) and then bring it down in 3 or 4 minutes to 700ish rpm. If it CANNOT bring it down by normal means, then many resort to something called FUEL CUT IDLE where it actually turns off the fuel momentarily and the RPM will drop because the car is stalling and running out of gas...then back on and it starts to rev back up and when it gets too hiugh it turns back off. Cars that do this can have annoying surges from 500-2000 back and forth at idle forever.

a vacuum leak or MAF meter problem can cause rough idle. ALmost any engine code that turns on CHECK ENGINE lamp may be suspect, does it have any current fault codes?

As well, consider Sticking manifold runners, EGR systems, and temperature sensors all can cause surging, so can a defective Oxygen sensor, slightly jumped timing, and dirty throttle plates the ECM wants to close more but they wont due to dirt. If its not something fairly obvious, it may require deeper checking.

probly shud du a data stream-shows values of system to help pinpoint prob--done w xpensiv gear at independent shops----or concentrate on throttl body--idle motor area or tps-sounds like somethin worn in the throttl opening causing it to almost close----some components cud be gettn hot causin hesistation-from circuit wear--unplug computer then plug bac in a few times-see if system reboots---haynes manual hav dtailed ngine repair-used by pros--or librarysec 629 has chilton books

With a carby the problem will occur because the engine is trying to work when you place it in drive and go nowhere ( same as putting a brake on the crankshaft). Use more choke or set the auto choke to run slightly faster. IE close the throttle butterfly more to increase engine speed. It all has to do with the fuel trying to evaporate in the carby and icing up the working bits and you need extra fuel to overcome this problem for a short term

if the car is running does it have the check engine light on? if yes what is the code if no then check to see if temp sensor is reading the right temp by plugging in a scanner that can read all data to give you an accurate reading... if it reads wrong temp it can cause running issues

I will suggest that you replace the fuel filter

Possible Idle air control, simple to change on the throttle body

i am not sure about the 2004 models.[i left dodge in early 2004] but all the previous years had a problem with running out of gas with 1/4 tank reading on gauge. it may be when you parked truck is sitting at a different angle than it was on the road, causing what fuel was in there to drain away from the pump. easy check is to crank engine a coupl of seconds then locate the shrader valve on the fuel rail on intake and depress the valve core [like letting air out of a tire] if air or mostly comes out under a little pressure then you are out of gas also i dont know your tank capacity but if all of a sudden your getting better mileage than before then that would indicate out of gas. a new fuel level sender is needed if out of gas.

Timing belt/chain may have been put it slightly off time, when motor rebuild

Check the fuel pressure at the shredder valve. You should have 60psi to 65psi with the fuel pump primed and engine not running. Once the engine has been started, there should be a 5psi drop. Make sure the fuel filter has been replaced and that the fuel pump is making at least 60psi. Anything less or more then 65psi will not activate the fuel injectors, the other thing to is that the spider injection system on the vortec are know to go bad and you may have to replace the system. Good luck and keep me posted, be glad to help you get your S-10 running 100% again. 

Your ECT (Engine Coolant Temp) sensor is bad. What's happening is at cold start it richens up the fuel mixture and as it warms up the ECM thinks it's still cold and essentially the "choke" is still on. Also check your coolant level. If you don't have enough coolant in you engine/radiator, the ECT doesn't know what to do so it thinks it's cold.