Thank you for using just answer. This vehicle is not equipped with a immobilizer system, however the alarm system can be activated and deactivated by using the key in the driver door. If the key will not unlock the driver door you would need to remove the interior panel to see if the lock rod is disconected/damaged. Has this vehicle ever been broken into or jimmied for a key lock out?
Can you elaborate on the idle issue???
Ok, now i'm with you, thanks. If you have no check engine light on and the vehicle is operating normally aside from this then most likely you are dealing with a clogged IAC (idle air control valve). If you just performed another repair and it started after that then I need to know that first.
I would start by checking to be sure that the air filter is clean and also remove the plastic intake tube that runs to the throttle body and then soak a clean rag in intake cleaner and scrub out the throttle body and throttle plate, some times the throttle flap can clog with carbon and cause this issue, if this does not correct the issue then you will need to replace the VERY expensive IAC valve that costs $606.02 or try to locate a used throttle body with a good IAC vavle.
The timing belt job would not really cause this unless he left a vaccum line loose and you would have an obvious hiss noise under the hood, you would also have a check engine light if this was the case.. I typo'd earlier, the IAC is not directly attached to the throttle body on this car, it is connected to the front of the intake manifold, the image above shows that. The image below shows the throttle body, start by cleaning this out as I indicated earlier
Former Lexus Master technician
Master Certified with lexus, 8 years experience with Lexus