Oboy. I don't know where to begin.
First, why was the exhaust system replaced? What part(s) was(were) changed? Do you know if they replaced the oxygen sensor when the did the exhaust?
The engine shaking was a cylinder not firing. The mechanic never fixed the problem. And, since the engine was shaking when you picked it up after the exhaust change, it makes me wonder if he used a "generic" exhaust system and eliminated/broke/did-not-check the operation of the upstream oxygen sensor, which if it's bad can make the engine shake. Changing the mounts was simply bogus.
I thinkcommunicating the problemto us may be compounded by your having misunderstood the dealership in Denver. A 2005 has an Idle Air Controller to control the throtte, operated by the ECU based on its input signals. One cannot "adjust the throttle" as in he days of old.
You didn't say why the exhaust system was replaced or how much was replaced. But of the engine was running well before that work was done, I'd highly suspect the upstream oxygen sensor. If the exhaust was changed to solve an operating problem, we'll need to know what that is before making any guesses.
One thing I can tell you for sure.....you need a new shop.