It's correct, but more natural-sounding examples spoken by an English person would be: "I like to keep everything in order." "I like everything tidy." "I like to stay tidy."

Yes its what I would say to someone, you are contrasting mess with order so there is no need to qualify it.

Yes, "order" is correct here. It has an uncountable sense. "Mess", on the other hand, sounds a bit like broken English to me. I'd expect it to be used as a countable noun, that is "a mess". But actually it seems more natural to contrast "order" with "chaos". Or, maybe, yeah, messiness - tidiness.

yes, I meant the room. Thank you Keith!

You have to tell us what the situation is. It depends.. (peope will say 'yes, it's ok' but they don't know what you're talking about!) 'I like order' seems like in society: order or chaos.. I don't think people would say 'order' if it's something simple like your room. They would say 'I like tidiness' in that case. Or 'I like them to be in order' if it's books or CDs etc..