I think you should investigate giving the car back to the dealer.
Yes, the radiator cap should have a spring. There is a plastic thingie that captivates the spring/gasket assembly, which break/fail when they get old.
You obviously overheated it. That's bad. 2JZ engines are tough, but NO engine likes an overheat. Hopefully (if you keep it) the engine wasn't hurt.
The "low input" trouble code is likely on account of either an unplugged ECT sensor, or a bad ECT sensor. It is located on the bottom, back-side, passenger-side of the radiator, and screws directly into the endtank of the radiator.
As an FYI on sellers resetting trouble codes before your test drive:
Most of the faults the ECU is designed to detect are "2-trip faults". The first time the ecu sees the fault, it sets an internal "malfunction", but does not illuminate the MIL/CEL. If you hook up an OBD scanner, you will see the malf. That malf stays stored in the ecu, and will eventually clear itself out if the failure condition is not repeated after so many trips (50, IIRC). Otherwise, if the malf is still stored in the ecu and the failure occurs a second time, the MIL is set, alerting the driver there is a failure condition.
At the very least, when you are looking at a used car, you should drive it for 15 minutes or so, shut it off, start it up and drive another 15 minutes. This gives you 2 trips, and the ECU the opportunity to illuminate the MIL. Most of the faults the ecu is designed to detect will be checked in 15 minutes of driving. A big exception to this are the evaporative emissions faults. Those usually take several days, and several opportunities for the car to fully cool off, before they post. There are a few other exceptions as well.
Even better, if you're a used-car person (like myself), is to invest the $100 in your own OBD scanner. It's so easy to check out a car you're interested in, there's really no excuse not to take the minute or two before and after driving and check it out. If you hook up to a car and find it has zero trips since the last reset - DANGER! Someone is trying to fool you!
This actually saved me $1000 on my IS300 purchase. I found zero trips since last reset and asked dealer about it. He said he "didn't know anything about that". Drove 10 minutes, shut off, drove another couple minutes and, lo and behold, the MIL came on. Checked for codes and sure enough, 2 of the O2 sensors were angry.
Last edited by Hodgdon Extreme; 05-24-2019 at 01:58 PM.