It may actually have something to do with the camshaft timing.

The p0022 is a bit concerning. If the cam phaser isn't turning enough to keep the engine in time, that can throw the p0345 also. Along with acting like it has a misfire.

 

So it seems to happen when he accelerates quickly, and it goes away when he revs it up slowly.
YouTube seems to believe that it could be a stretched timing belt and perhaps a busted timing chain guide and/or phaser gear (though I think the latter would cause a persistent, rather than periodic failure). These seem pretty plausible as I can see a slack chain getting out of time with a hard rev followed by a quick slack. Parts don't seem horrifically expensive and the labor seems doable, though time consuming. I think we have time if he babies it. The engine has 180,000 miles on it, and so I think a cheap replacement would be justified if we plan on swapping the motor in the next year or two.

 

I would make sure I have all the misfires fixed first.

You need a better scanner. Or maybe forscan.

 

if it is sluggish it could also be a catalytic converter clogging up. They will cause a misfire as well.

That phaser is ran by oil pressure. So that is something to think about too.

 

P0316 (cyl 6) is usually an ignition coil or coil circuit failure. This is different than a p0306 that more of a generic misfire code. Swap the coil from cyl 6 and another cylinder. See if it follows the coil.

P0345 was a common code on 05-08 gt with an alternator that had a failing rectifier (diodes). They created electrical noise that messed with the sine wave of the DC voltage and confused the PCM , thus it threw the code. An alternator can charge just fine but still have a failing rectifier.

Start with p0316 first. It's possible that a failing ignition coil or circuit can do the same thing as a bad rectifier as It may create electrical interference

 

Are you sure about that? The P0316 says "Misfire detected during start-up - first 1000 revolutions".

 

Apologies , a P0356 would be a coil circuit failure.related to cyl. 6.

P0316 is indeed as you've described.