Remote Start, will start the car and in 5 seconds tunrs off.

if you have remote start dealer needs to flash it

Chevy Impala 2000 2005 Factory Service Workshop repair manual

for the easy solution download the pdf file

Did you reprogram the remotes if so there is a modual in the car that could have gone bad this seem to be a common problem on impalas for some reason not sure why

Does it happen when you while/if driving, when you enter your car and close the door or when you just remote start it then stay outside your car?

Is this a factory option remote start or after market?
If after market, the passkey bypass module may be malfunctioning which, after a few attempt at starting the car, will cause the car to go into tamper mode and prevent starting via remote start or key for a few minutes (10 minutes to an hour)

this is normal operation for gm factory remote start if the ck engine light is on the remote start will be disabled. Because with the ck engine light means that there is something not operating properly and gm has put this in there system so you will not damage the engine by remote starting the vech with low coolant and let it run for 10 min and over heat the engine and cause engine damage.

if thats a key resisor bypass i never heard of that, i would check with with bulldog to see if may be a set upproblem

you should be able to however im not sure if it is associated with the on board system if all else fails reconnect it and maby change the batteries in the remote for a starting point

ECU fault ,dealer only

Check your manual that came with the car, there should be a reprogamming sequence in there. Otherwise, is the car still under warranty? Go to the dealer and have them check it out. It may be the receiver on the car that is malfunctioning. Do you have a second set of keys? Does that set work correctly?