Hi, I was using a dairy coffee mug that was washed and clean (although, it still has those coffee stains inside it, and it was used in the last 24 hours for coffee with milk) and so I put the pareve coffee mix inside the mug and fill it up with hot water from the hot water machine, but the spout of the hot water machine is sitting below the brim of the mug and the coffee from the mug overflows a little into the spout. Would that change the status of the spout from pareve to dairy and then need to be kashered?
No, the spout remains parev, and there is no need to have it kashered.
This transfer of taste is known as nat bar nat – there is taste in the cup, and this is transferred by another substance. It does not have the power to make another utensil dairy.