In proper use they are synonyms. The word חירות is more formal (like in חג החירות) or legislative (like in חירויות הפרט which are individual's "freedoms" or like in חוק היסוד כבוד האדם וחירותו which is a basis law about the respect and freedom every person diserves. You don't have to remember this because it is a matter of time until the current government will cancel it).
Yes, it is.
But you should be aware that the word חופש, unlike חירות, is widely misused as holiday/vacation instead of חופשה, like in החופש הגדול which is the Hebrew phrase for the July-August school break - "the long holiday" - or חופש פסח which is the 2.5 weeks school break surrounding passover.