Can one learn to play an instrument as naturally as they can sing a song from recall?

Yup, it's just improv. Most commonly you'll see pianists being able to do it quite easily.

Yes. That's the goal of learning musical improvisation. I can do it on drums for most stuff, but I'm relatively new to the world of pitched instruments, so I'm not there yet.

If you mean hearing a song and being able to roughly recreate it, yes, with years of training