The circuit you hear powering up for two - five seconds should be the fuel pump priming the fuel system . What type transmission ? manual or automatic ? Do you know what a wiring diagram is an how to read it ? Free wiring diagram's here http://www.bbbind.com/free_tsb.html Enter vehicle info. Year , make , model and engine size. Under system click on engine then under subsystem click on starting .Click the search button then the blue link . If you look at this diagram you will see the fuse at the top of the page that powers the ignition switch . You will see the ignition switch , the starter relay and the PCM - engine computer ! The PCM supply's a ground to energize the starter relay . But only if it see's that the transmission is in park or neutral ( automatic trans ) Or the clutch switch is closed ( manual trans ) .To see this you would need to go to transmission controls under transmission/transaxel . Plus the PCM will not energize the relay if there is a anti-theft system problem . You could do testing right at the relay base (where the relay plug's into ) With a test light ,voltmeter etc... Watch this video on relays
Your circuit is analog. See the upper right corner of the diagram. The PCM is getting the signal, or the car would not be shifting right. You must have a bad connection from the "T" in the diagram to the cluster. If that "T" is under the car, it is susceptible to the harsh environment.
I have seen fuel pressure issues cause this. Fuel filter, or fuel pump. However, I have also seen the catalytic converter cause this issue. The honeycomb, located inside the catalytic converter, breaks up and will block the exhaust when you accelerate. Engine power is lost, until you let off the gas and let it idle. Then, the chunk falls down from the exhaust port, and it runs fine again. Until you get on the gas and increase the velocity of the exhaust, and the chunk blocks the exhaust gasses again. Good luck
The dealers are just telling you thing's it could be , doesn't mean they are just going to replace all of those parts or you need all of them , Doing the diagnostic test procedures associated with the DTC - diagnostic trouble code are were they start . Take it to one an have it diagnosed .