Weird Idle/Power Issue

If this were my car I would be worried about one or more weak grounds. The cluster is grounded at G204 which is located behind the center console. G204 is shared with many devices including the GEM and Radio. It's frequently messed up during a radio install.

However given your symptoms the weak ground could be anyway. In the trunk or right at the battery.

Here's is some information on how to run a voltage drop test. If you understand the "concept" this test can be used to locate weak ground no matter where it is.

Howto perform charging system voltage drop test

Suggest starting with the battery grounds and the grounds around the radiator core support. Are they CLEAN and tight?

Is the motor grounding strap in place between the left hand motor mount and frame rail?

At the very least start with a through review of the battery, battery cables, and alternator.