Lurches foward when changing gears and revs engine by itself while changing gears and when slowing to stop

Based on the symptoms you have described, it sounds like there may be an issue with the transmission in your Mazda 3. The loud clicking sound when shifting into reverse could be a sign of worn or damaged gears, while the hesitation and lurching in first gear could be due to a slipping or malfunctioning clutch.
It's also possible that there could be an issue with the transmission fluid, such as low or dirty fluid, which can cause shifting problems and other issues. I would recommend having a qualified mechanic inspect your vehicle to diagnose the exact cause of the problem and recommend the appropriate repairs.

Transmission problems. Have the codes read. Most problems with this is internal transmission. Consider the cost and the vehicle.

that is a hum dinger, largest ever posted, in one shot.
very good.!
one word , the transmission slips. (never drive a slipping tranny)
overheat indicator.! that is key.
many newer cars will change the transmission shifting any time
the engine over heats.
and overheat wins over transmission , every time.
if the engine burns up, what good is any transmission.? nada.

solve overheat first.

slip or flare.
is when the engine RPM rises fast and the car dont go faster.
SLIPS.
and slipping burns up trannyclutchs and increases the damage. inside the transmission, dont do that. dont allow it,
park it and fix it.

on some tranny there is a temperature sensor that the TCM
watches and changes shift patterns based on AFT fluid temps.
and if the ATF fluid gets too hot , it takes actions. to keep the tranny from harm.

why not get a scan too and see what DTC errors there are..
DTC, engine overheate.
DTC< tranny overheat (caused by above)

The gearbox oil and filter or screen

Stop driving it and check the trans. fluid level. When they pulled the shaft they lost fluid out the tail shaft and forgot to refill it. I hope the trans. is not damaged, good luck.

sounds to me like you need to get your auto transmission serviced

ignition modules can cause problems like this also airflow meters and faulty fuel pumps

Sounds like the torque converter isn't unlocking quick enough. Drive at 45mph,keep foot on gas and use other foot to touch the brake and watch the tach to see if it speeds up,when converter releases. Check trans fluid level hot.If it is more than a couple years old,change. Use the correct fluid for Toyota and add a bottle of Lubgard. Have the throttle positon senor scanned and check any idle control circuits. Throttle Body may need cleaning.

the engine tickover is too slow when the engine is cold its running on cold start, ie:choke,, after it warms up the engine slows down this is what is causing it to stall in gear,, get some one to hold the brakes hard on and with the engine running in gear ajust the tick over till its reving at 750 - 850 rpm the revs will go up a bit when its out of gear,, before you pick a gear to go into push the foot brake down to stop the car lurching,,,

Possible motor mount worn. Stop at a reputable garage, they should be able to confirm for you at a glance.