Car does not start

Lets start with the main problem that happened which is that the car out of nothing just fully turned off (engine off, dash lights off, everything was off). When this occurred there was this burning wire smell outside the car. I instantly though of a burning wire or something and left it. After a couple of min the car lights came back to live, but when i tried to start it, it didn't start. I could hear some crank and eventually it stopped fully.

After diagnosing the car i could not find any burning wire which was very strange.
- Firstly tried jump starting the car with jumping cables which was no success and i could hear a crank, but no start. After this the smell came again.
- I checked the voltage of the battery and this was above 12. When i tried to crank it, it then drained to a lower voltage. After this some friends of mine pushed the car and jump started it which eventually did work and i checked the battery again to see whether the alternator was working. It gave me a voltage of 13.6 which indicates that it is working.
- My next thought was that it might be a starter issue, so i took the whole top out in order to get to the starter. After this i checked the starter and this seemed to be working fine, which confused me even more.
- The only way the car can start is by push starting it. Also when i sometimes try to start it, the fan kicks in while the engine is cold and never been on which makes it very strange.
- All fuses are good and checked all the relays. Tried to check for error codes on the computer and could not find any.
- All terminals are cleaned and corrosion free.

I am so confused at this stage and do not know what to check next. I thought about buying a new battery, but wouldn't a jump start cable make my car turn on?