Water from a plugged-in urn

Hi Rabbi. If water is dispensed directly from urn into a cup, is that cup (and the water inside it) a kli shayni (for kashrut and for shabbat) even though the urn stay plugged in when water is dispensed?

Yes, the cup is a klei shaini.

Hi Rabbi. Likewise, am I correct that if spout of a hot plugged-in urn touched a (probably dry) part of inside of a dairy-side mug (with that mug containing some hot liquid), that no taste-transfer occurred? (For example, correct that this did not at all make the urn dairy?) Thank you Rabbi.

No issue