That's got duovalve written all over it. I had the exact same thing happen on my C280. Your coolant pump code relates to the small pump on the right fenderwell that circulates coolant when you press the 'rest' button. It also assists the heater circuit at idle, othewise the duovalve opens to compensate for the low flow at idle, and when you take off, you get a blast of hot air until it can compensate. The sensor problem is one I'd have fixed at the same time.
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'96 C280 (gone)
'97 C36
'05 C230k
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