First step - have the check engine light or stored fault codes scanned - for free - at any local auto parts store; Advance Auto, Auto Zone, NAPA. Then, post a reply as 'add answer' for more assistance.

Possible Causes of Engine Hesitation or Stumble:

Dirty fuel injectors (cleaning the injectors often fixes this)
Dirty air filter (change filter)
Dirty / clogged fuel filter (change filter)
Bad MAP (manifold absolute pressure) sensor
Bad TPS (throttle position) sensor
Bad or dirty MAF (mass airflow) sensor
Low fuel pressure (leaky fuel pressure regulator or weak fuel pump)
Vacuum leaks (intake manifold, vacuum hoses, throttle body, EGR valve)
Bad gasoline (fuel contaminated with water or too much alcohol)

Sometimes, what feels like a hesitation is actually ignition misfire rather than lean misfire. The causes of ignition misfire may include:

Dirty or worn spark plugs
Bad plug wires
Weak ignition coil
Wet plug wires

If you are asking how to replace it, the air induction system and air filter box need to be removed to access the external filter.

Engine mounts - figure out which one or replace all.

First step is to get a second opinion and detailed diagnosis from a trusted and reputable repair shop in your area. There is a possibility the transmission slipping just coincided with the CV / axle replacement.
However, to answer your question - there is no reason to believe the CV joint replacement may have caused the slipping - unrelated parts - except the transmission drives the CV joint and front axle.

UNLESS the drive axle that was replaced was not fully seated into the transmission and the noise you hear is the splines slipping against it's counter-gear. The other possible cause is that when they removed the CV axle, maybe too much transmission oil was lost (it leaks out the side of the tranny while the axle is removed) and they did not replenish it. Either way, this really does not seem right. They should have asked you to return asap to check it instead of trying to prep you for another $2000 bill for a tranny you prob do not need. If I were you, I would get a second opinion from another shop before going back to that shady place. At least you will know the real cause. The dealer might just lie to you and sell you a tranny. Do not take it to another dealer so there is no biased.

my 5 spd would hang up in reverse ,replaced top motor mount, works fine,check your motor & trans mounts.

What is throwing me off here is trying to find a common cause for both the bucking and the door locks cycling. They both have causes on their own, but I can see no relation to both. Are you certain that both conditions began at the same time / the same moment? The locks could be a bad Power Door Lock Switch on the Driver Side Front Door. Could also be caused by a bad Body Control Module. Try unplugging the electrical connector from the driver side power door lock switch and see if the locks stop cycling and if so replace the switch. I would not think that could cause bucking but heh, maybe the electrical load is doin something. Find the cause of the lock problem and see if the other problem resolves.

Plug a code reader into the OBD connector under the driver side dash and read fault codes. It takes like 3 minutes and a $20 code reader from Amazon can do it just fine. Update the post with code numbers please. This is how cars get diagnosed.

700 just means that there is a transmission fault, the fault in your case is 1810 which may be a bad TFP Valve Position Switch. This is inside the unit.
128 means that *most likely* (probability greater than 90%) your thermostat is stuck open causing the engine temperature to remain too cool, so change the stat. The 455 may be a bad gas cap. the 127 is an error you made reading the code because there is no valid code P0127 for a 2003 Saturn L300. Erase all the codes and wait to see which ones return.

Please rewrite this with additional details. As written I do not understand it at all. Explain what you mean by saying it will not downshift? You are saying that when you come to a stop and take off again, that the transmission is stuck in drive? And the other thing, you are saying it will not crank?