I shave every day, but my beard stays the same. What am I?

I am first found in caves, now prolific online; I am a depiction, a drawing, a symbol, or sign. I will convey whichever mood you could wish; or for that matter, a fist, flask, or fish. What am I?

I welcome the day with a show of light, I stealthily came here in the night. I bathe the earthy stuff at dawn, But by noon, alas! I'm gone.

It's pretty hard to give up.
If you remove a part of it, you will be left with a bit.
Even if you remove another part, the bit still remains.
Remove one more and it still remains.

What is it?

When you say goodbye, I am what you need to tie.
I come as a couple and go wherever you run.
Who am I?

Who has married many women but was never married?

I was born in a small house and I live in it all alone. No windows and no doors have been assigned to my house. The only way I can go out is by breaking through the walls of my house.

Who am I?

Two in a corner, one in a room, zero in a house, but one in a shelter. What am I?

I have cities, but no houses. I have mountains, but no trees. I have water, but no fish. What am I?

A finger goes in me. You fiddle with me when you're bored. The best man always has me first. What am I?