Yes. Comedy is a literary or theatrical genre. Tragedy, for instance, is another. For example, Shakespeare wrote comedies, which were amusing, tragedies which were dramatic/serious, and historical plays, which were mainly factual. Humor (or humour) is a human faculty. If a person has a sense of humor/humour, they have the ability to find situations amusing. As you can see, these words are used in different contexts. However, if you are searching for different kinds of films, for example, the 'funny' ones might be in a section called either Comedy (i.e. they are comedy films) or Humor/Humour (i.e. that they will appeal to people's sense of humour). In this situation, the two words might be used to mean the same thing.

Yes, but it is difficult to explain and there is a lot of overlap. Each word has specific uses and "colocations." I think I will just give examples, as I can't give a coherent explanation. Someone with "a sense of humor" is someone who sees the funny side of things and makes jokes. Someone who stand up in front of a microphone and tells jokes is, properly, a "comic." However, her art form is "stand-up comedy." Comics are often called "comedians." A play, move, or TV show that has a happy ending and makes you feel good is a "comedy." Usually, a "comedian" means an actor who specializes in comedy, either full-length plays or short "sketches." A "comedian" can also mean a playwright like Neil Simon who writes comedies. An author like Mark Twain who writes in an amusing way with a sense of humor is a "humorist."