A dirty throttle body is a classic cause of a hunting or variable idle. You might give that a cleaning too while you're in the engine compartment.
A secondary air system leak might be the cause of the on again and off again CEL. it comes on in cold temperature conditions. If you're in FL, it might not start the secondary air pump. If enough good cycles go by, the CEL might think everything is working again.
All that said, I'd start along the lines you're planning. Clean everything. Clear all the codes and see what happens. If the idle is steady, the one problem solved. If you get a CEL again with an aging related code, the replace the aging part. Then get the secondary system leak tested.
Best of luck
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