Where is the ECM located on a 2003 SC430? Are they plug and play?

 

The main ECM is located under the hood, behind the left front headlamp and windshield washer fill neck.

 

Can ECM be tested? Is it possible that the ECM is stuck with a certain code?

 

Tested? Not without the Lexus DTC Scan tool as far as I am aware. I looked through my 2002 Repair manual and did not find a P0051 error code so I cannot help there. But you may be able to force a manual reset by disconnecting the battery for 30 minutes. Doing so will cause the loss of radio and seat memory preset. Both are covered in your owner's manual.
Regarding the VSC lights, there's been lots of discussions about them in the past. The search and ADVANCED seach tools at the top can be very helpful. I found this link that covers a "jumper solution" for resetting the VSC lights. https://www.clublexus.com/forums/sc-...scangauge.html

Oh BTW welcome to the forum.