I need to know if a 2002 ls300 drivers side door will fit a 2001 saturn ls200

A salvage yard would be a likely source of interchangeability information.

https://www.autogenius.info/saturn-vue-2001-2004-fuse-box-diagram/
The fuse box diagram above says you have a drivers side relay and passenger side relay. If that is correct you might try change the two relay. If the passenger side does not work after that you know the relay is bad. If the two relay are the same. It also show a relay for all the doors.

https://www.2carpros.com/questions/saturn-ion-2004-saturn-ion-power-door-lock

https://www.justanswer.com/saturn/6d6ac-saturn-ls200-reset-anti-theft-system-engine-shutdown.html Look this up and read will work. 30 minute process.

Check coolant level...may just be low on anti-freeze ...next check would be the thermostat...hope this helps.

Call your nearest automotive body shop and they can tell you. They might even have one on hand, or sell you one at a discounted price. But first make sure you have the VIN.

I need more information. When the radio is on, does the display show 4 lines like this. ---- or does it spell the word "lock" on the display. If it says "lock" , you must take it to a G.M. dealer .
If it shows the dash lines, then turn off the radio.Turn the key to "run". Hold the preset buttons 5 & 6 down for 5 seconds or so until you see the ----. Using the seek /scan button, enter 0000.If it doesn't unlock, then some other 4 digit code was previously entered. Off to G.M. you must go.

Charge battery, if not, replace.

mcdevito75 here, Sounds like the linkage on the inside of the door came off the handle, let a glass repair shop fix it.

check fuse also check for disconnected wire under the seat

2001, 2002. Be sure to bring the computer and associated wiring harnesses.