The word "chilling" usually mean your at a relaxed state. For Example, if you would ask me, "what's up" = How are you. I would answer like this, "Nothing. Just chilling." Meaning I am not really doing anything of significance, just letting the time past by. Whereas "Hanging Out" implies that I am doing something usually with other people. So if I say, "I am hanging out at the mall." Its taken as you are "proactive" or doing something. But if I say I'm chilling at the mall. It can be taken as I'm just at the mall not really do much. I am chilling at home. ( You are at home not really doing anything, maybe your even bored.) I am chilling out home. ( Doesn't really make any sense.) I am hanging out "AT" home. ( When you say this usually someone is with you and you are doing something like video games. I am hanging around home. ( This makes sense but it usually isn't said. It means your just at your house or around it probably doing nothing.) REMEBER: Chilling is a slang word.

To answer the last question: You should say "I am chilling at home." Because this phrase is so informal, you may want to use the contraction for "I am" = I'm. "I'm chilling at home." Furthermore, "I'm chillin' at home." is even more natural.

I am chilling at home = I'm at home relaxing, not doing much I am chilling out home isn't an actual phrase/ sentence. I am hanging out at home is just staying at home, basically the same as chilling. I am hanging out around home is just at home, doing random things for the most part I don't really understand what you mean with the last sentence :P