Sounds one octave higher than should be

If I use a separate tuner, outside the computer, the tuner shows C5 when I play a C4 note. (A MIDI Monitor plugin on the tracks verifies that it receives C4 from the keyboard.)

This happens for several libraries I’ve tried: Iconica Sketch, BBCSO Discovery and some PianoBook samples. The only fix I’ve found so far is to adjust the instruments themselves. For example, by transposing the Iconica sketch patches -12 semitones.

Is that normal to have to do? Is there some other simpler setting in Cubase that fixes this?

That is normal. Different manufactures have different middle C. Some use C3 some use C4.