I have a problem with my car engine; it stops working as soon as I start it.

First and foremost, have you tried scanning for any active codes with a scan tool? That would be my first question, as otherwise, this will help us determine which way to go instead of throwing parts at it.

As far as the crank sensor, you state a 2.0 engine, however on my end it shows a 2.4 (4 cylinder) or 2.7 (6 cylinder) as available options, so can you please confirm the engine for us.

Believe the 2.0 Theta engine was sold only in Korea in 2004 and the US got the larger brother in the 2.4 engine. Pretty much the same engine other than the displacement. Missing at high rpm and then getting worse until it's missing at lower speeds and now doesn't want to start sounds like it is running lean. Does the CEL light come on at all? If yes you would start there as Kraken said. If not I would look at the EGR valve, if it is binding and not closing fully it can cause these symptoms. It can let extra air into the engine at initial start and prevent it from starting and at other times it may be sticking so the pintle doesn't seat and dilutes the fuel charge enough that it misfires. The 2.4 has a known issue for this where the control solenoid fails or the EGR valve itself sticks. A quick test is to pull the vacuum line off of the EGR valve and plug the hose you removed. See if it starts and runs now. If it does, remove the plug and start the engine, do you feel vacuum in the hose? If yes, change the EGR vacuum control solenoid. It looks like the first attached image and connects with the EGR valve in the second image.