That's the PCV and all (looks like all) the vent hoses. I suppose with the car's age and mileage, it's not a bad plan to replace all of these items. And if the PCV is blocked it could cause some issues, but I am not sure if it will cause the problems that you have been struggling with.
I am hoping other member will add to this and add their thoughts.
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BMW E90 330i (M) '05 (His) | (Hers) BMW E46 320i f/l (M) '03
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