I’m the part of the bird that’s not in the sky

I’m the part of the bird that’s not in the sky. I can swim in the ocean and yet remain dry. What am I?
A shadow.

I am mother and father, but never birth or nurse. I’m rarely still, but I never wander. What am I?
A tree.

I went into the woods and got it. I sat down to seek it. I brought it home with me because I couldn’t find it. What is it?
A splinter.

I am weightless, but you can see me. Put me in a bucket, and I’ll make it lighter. What am I?
A hole.