Is there any objective evidence that different keys

1. Do different keys have different moods?
2. Why do different keys have different moods?
3. Why do composers use different keys?
4. Do different keys sound different?

Do different keys have different moods?

These were always a bit of a convention, created through associations that built up over decades of composers using certain keys for certain purposes. There's nothing inherent in the sound of different keys that conveys those specific moods.

Why do different keys have different moods?

Keys do not set a mood, but rather, the combination of chords and chord shapes, for instance when you go from an Dm to Am most people would say that it sounds sad. You can play this anywhere with the same chord shape and it will have the same kind of sound.

Why do composers use different keys?

Music for solo voice and piano is often made available in different keys to suit the voice range of different singers. You would notice if the same singer sang a song in different keys because it would lie differently within the voice range.

Do different keys sound different?

So the short story is that different keys are / were fundamentally different due to the application of ratios not preserving the exact frequency values of notes. This is apart from the the sound of higher/lower frequencies, which of course affect the aesthetics as well.