When transmission fluid smells burnt, Does that mean that the transmission is already bad?

Maybe dirty sticking parts. Does fluid on stick look dirty or smell burnt ? Try tranny cleaner like Seafoam Tranny Tune. After driving with that about 1/2-1 hr, get tranny fluid changed. Filter inside likely fine. So if issue getting pan gasket, a place that does flush via dip stick tube will do.

Rebuild the trans. Kit on RockAuto.com

That is one indicator, but I would change the transmission filter and fluid

MANUAL TRANSMISSION, CLUTCH SLIPPING
AUTO TRANS. CHECK FLUID, IF LOW ADD
CORRECT TYPE OF FLUID, IF BURNT SMELL
HAVE DEALER CHECK

Have you checked the fluid level?Was there any oil leak spots under the car?
Correct any low fluid level and give it a try.
If the fluid is at the correct level,disconnect the battery for 10 mins then reconnect and try it again,if it still won't work you may have to have a transmission expert take a look at it.
You can put in a used transmission if needed and not spend lots of $$$ for a rebuilt trans,or just buy another complete vehicle to drive and not fix your 2002.

Check the transmission fluid level. This should be done after the car has been driven long enough to get hot air from the heater - this means the engine has warmed up. Do not simply let the engine idle until warm - it should be driven. Park on a level surface and with the engine running and brake set, locate and remove the transmission (do not confuse with oil) dipstick. Wipe the entire length of the dipstick with a clean rag or paper towel, and return it fully to the holder tube. Look at the color of the fluid - it should be pink - not brown. Smell the fluid on the rag / towel. An oily smelly is expected but it should not smell burnt. Often, brown fluid and a burnt smell go hand in hand. Remove the dipstick again and check the level. It should not be low nor over-filled. Add fluid if low.

If the fluid is at the correct level, or smells burnt or is other than a pink color, you may need to perform a potentially expensive repair / rebuild of the transmission. You *may* solve the problem by changing the transmission fluid and filter, but often times it *may* only buy some additional time before the work needs to be done.

Ultimately, the car should be brought to a trusted mechanic - or transmission repair shop for evaluation / estimate. Transmission work is a specialized line of work. Most auto mechanics do not do this work (including many dealers) and send it out to these specialty shops instead. You can save a few bucks by going to the specialty shop directly and not be subjected to the mark up the auto mechanic adds to their bill.

I hope this helps and good luck!

This would mean (a) the transmission fluid temp sensor is bad, or (b) the computer is bad, or
(c) the connector or wires to the sensor have corroded/been damaged somehow.

Definition of your code:
P0712 Transmission Fluid Temperature Sensor Circuit Low Input

which means that the voltage coming from the sensor is read by the computer as too low,
which also means that it could be zero. Zero voltage can happen if the transmission fluid temperature
sensor connector has been unpluged, or one of the wires to the sensor have been cut/damaged/corroded.
If there has been recent repair activity to the car, then its possible that this sensor's wires or
connector has been disconnected/cut/damaged somehow.

To fill the transmission fluid, you need a funnel that fits into the dipstick tube, add fluid there. If the fluid is black and smells burnt, that's bad. I would have the fluid changed. These transmissions don't have a filter, probally a bad idea.

sounds like you have a bad heater core leaking or seaping radiator fluid into the air ducts

If transmission fluid ran more than 25K miles it neds a change,some people think it doesent need to be changed and is not thrue.