Watch some Jacob Collier, he often changes tunings mid-song (i.e. on Hideaway, he starts with A432 and ends at A440 (?), and in In The Bleak Midwinter, he modulates mid-song to G-half-sharp, or A453 iirc). This is just mind blowing when you notice it, like it's a very detailed painting which shifts slightly right before your eyes, and the shapes are still familiar, but if you squint, you can see some weird colors
And he talks about it extensively in master classes, fascinating stuff
As for songs in wrong keys, for me they just sound kind of wrong(-ish). Like, correct, but not quite at the right place maybe?