please help peugeot automatic?

In september 2012, I bought a 2001 peugeuot 206 lx 1.4 automatic, from a private dealer with no warranty and after a couple of weeks I noticed that there was always a strong burning smell coming from the bonnet, although it never felt that it got too hot and i couldnt really tell where exactly it was coming from, also when i was driving for the first little while the car over revs, to about 40,000 but only sits on about 30mph (it would prob rev more but the noise worries me and i ease my foot off accelerator) then the car will make a clunk noise n jolt forward slightly, that will happen 3/4 times and then do it one time massively, the car will then make 3 beeps with no lights flashing and then drive normally. I had the engine oil changed and a new filter put in place as it was filthy, hoping this would solve it but it hasn't,
Then i was driving home one night when all the sudden the temparature needle went straight from cold to hot and flashed stopped on the dash, i pulled over got out but the engine was not hot, i called out green flag who looked it over and said he didnt think it was a major problem he thought i probably needed gearbox oil but i had a slight leek from sensor swicth underneath bonnet which was coming from exhaust pipes to engine? and it was cheap to fix but should be replaced.
with this i went to my local garage,explained what had happened they ran a diagnostics and brought up two fault codes:
P1544 STEPPER MOTOR CONTROL
P0116 COOLANT TEMP SENSOR CIRCUIT
but he said this would be a cheap fix. he told me to run it for a few days, come back and they would run the test again for free of charge, so I done this.
When I went back to the same garage I saw a different man, I explained to him and he said that it sounded like a problem with my gearbox and if thats the case its best for me to get rid of it he said it could possibly be gearbox oil but he highly doubts thats what is making the car over rev and jump forward like that, sounds like its had it.
But Im not sure if that seems right?? Is it really that serious of a problem?
would I not have a light on the dash? Would it not do it the whole time whilst driving?
He ran the diagnostics and said it brought up the coolant fault code and I said but if there was a problem with the gearbox would it not show, so he said hang on let me check something, he tapped something in again and brought up this;
2 INCORRECT TRANSMISSION RATIO/SLIP
0 KICK DOWN SWITCH
The car is now booked to go back on Monday to have the gearbox oil checked and maybe replaced.
I don't know anything about cars, this is my first car and I am still on a provisional license. I dont want to be caught out, but I will have no idea what anything means or if what they are saying is right.
anyone got any ideas, as they arent automatic specialists i dont have one nearby and it seems like guess work to me. please help. thanks