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It sounds to me like they just messed up the timing. Setting the timing on these can be a bit tricky. I'd be surprised if the oil pump is the culprit.
In my experience the OCVs throw codes if you hit the VVT at a higher rpm but otherwise wouldn't cause a rough idle just on an initial start.
As for paying/not paying, that's between what kind of deal you have with the shop and their policies. Better to work something out with them. If you don't pay they'll eventually take the car via a mechanics lien probably. But that can be lengthy and painful for everyone.
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