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I'm gonna guess that cyl2 misfire has something to do with your crappy idle. Check the spark plug for proper gapping, and also check it for evidence of a mixture problem (which could be caused by the lack of 02 and coolant temp sensor, as the malfunction of these sensors will make the engine stay in closed loop all the time, making the mixture far too rich). I wouldn't be surprised if you end up replacing the spark plug due to fouling.
If I had to hazard a wild guess, it's the coolant sensor's short that caused the engine to go into closed loop mode, which dumped a shitload of fuel, fouling the spark plug, which then made the cylinder misfire, sending unburned/partially burned fuel into the exhaust manifold to foul the o2 sensor.
And Caveman, you're describing a classic timing belt break. You should not need an entirely new engine, just new valves, and possibly a new head if the damage is bad enough. Check into that - it's remotely possible the shop recommended a new engine because they can get it cheaper than a new head -- I doubt it, but who knows.
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