What Am I Riddles

Question: I can run but never walk. Wherever I go, thought are close behind me.

What am I?

Question: What has classes but moves?

Question: If you're looking for something sweet I know what to do.
But if you don't like heat I'm not for you.

What am I?

Question: I am a nice word composed of five letters, but the Romans would call me a numbers.

What am I?

Question: I go into the water red and come out black.

What am I?