Car is stalling and jerking randomly

Question edited for clarity and typos.
Question moved to model category.

They are a heap and generally disliked. Try cleaning the Throttle Body and the MAF (mass air flow sensor). It also sounds like a new exhaust would be a good idea. They make a difference to the engine management if holed.

https://www.google.com/search?q=2007+chrysler+pt+cruiser+clean+throttle+body+and+MAF

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Sounds like a gas tank full of crud. Check gas filter to see if it is dirty. Also water in gas will clog gas tank filter then when car stops water will disburse and collect when engine starts again.

There should be a trouble code or codes stored in the engine computer. A hand held scanner could retrieve them-go to Autozone for a free scan service. The codes indicate to a mechanic where to start looking-whether it is a fuel, or air, or ignition spark delivery problem.
Things that should be checked are the air filter, fuel filter, spark plugs, check fuel pressure, check for air obstructions into the engine (like a dirty throttle body inside-sometimes a cause of stalling), check for vacuum or air leaks that let unmetered air into the engine. There is an ideal ratio of gas to air mixture that the computer is always trying to achieve. When it does, the engine runs fine and gas mileage is optimum. This is why it is important to keep up with regular maintenance and service. Get the codes to see if the computer is doing what it does best, better even than an old mechanic with a good ear.

I would replace the fuel filter, double check your work and make sure that all plug wires are securely connected and going to the right place (we all make mistakes sometimes) and check or replace the coil.

ENGINE HESITATES WHILE ACCELERATION

LOSS OF ACCELERATION

It can be fuel pressure issue or it's the problem with accelerator pedal or its clogged filters causing there.

There are many other possibilities. I suggest you to go through the list of help links mentioned below, these will help you to confirm where the actual fault lies: -------

Jerks and Hesitates During Acceleration?

http://technoanswers.blogspot.in/2012/07/2004-dodge-town-country-jerks-and.html

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Loss of acceleration?

http://technoanswers.blogspot.in/2012/02/loss-of-acceleration.html

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Headlights flickering and car hesitates to start?

http://howtobyme.blogspot.in/2012/02/headlights-flickering-and-car-hesitates.html

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How exactly the Vehicles Accelerator Cable works?

http://repairhelpcenter.blogspot.com/2011/11/how-exactly-vehicles-accelerator-cable.html

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How exactly Vehicles Accelerator Pump Diaphragm Works?

http://repairhelpcenter.blogspot.com/2011/11/how-exactly-vehicles-accelerator-pump.html

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2000 Toyota Avalon: Engine Lean Codes and Knock Sound on Acceleration?

http://whatisbyme.blogspot.in/2012/05/2000-toyota-avalon-engine-lean-codes.html

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2004 Toyota Avensis: Acceleration And Deceleration makes noise on Front End?

http://whatisbyme.blogspot.in/2012/05/2004-toyota-avensis-acceleration-and.html

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Wheel hop under hard acceleration?

http://technoanswers.blogspot.in/2012/04/wheel-hop-under-hard-acceleration.html

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2009 Chevy impala accelerator pedal sensor?

http://technoanswers.blogspot.in/2012/02/2009-chevy-impala-accelerator-pedal.html

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Nissan Pathfinder stalls and bucks on Acceleration?

http://technoanswers.blogspot.in/2012/02/nissan-pathfinder-stalls-and-bucks-on.html

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Acceleration lag problem?

http://repairhelpcenter.blogspot.com/2012/01/acceleration-lag-problem.html

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These details will help.

Thanks.

Sounds like a problem with the fuel system.. Not 100% sure though.

these cars are carburated motors and need around 10 minutes to warm up because they operate at higher temperatures.

These is because of the improper combustion, so you better check the air filter and the exhaust along with that you replace all the spark plugs

I need to get a clear mental picture of what exactly is happening.

With this said, you stated that the car "runs fine" until warmed up.

Q: When you start the car when cold it idles fine?
Q: If you allow the car to idle until "fully" warmed does the car stall?
Q: As the car begins to warm up, Does the RPM fluctuate abnormally?
Q: Does the car Only stall when you press on the gas pedal or does it stall just sitting and idling in addition to pressing the gas pedal?
Q: If you can drive it while cold does the car stall during acceleration when warm or only at Stop Signs / Lights?
Q: If you wait 20min or so, can you start the car again?
Q: How many miles are on the car?

Possible dirty fuel filter. I'd start there first, as it's the cheapest route, and quite possibly replacing the filter will solve your problem.