I think the difference may be the heaven of other religions versus the heaven of Shintou, but I'm not 100% on it either way. Am I right, or is there still something I'm missing on this one?
Generally 天国 refers to the Christian heaven (while 地獄 can refer to both, the Christian and Buddhism hell). Both words are frequently used in everyday language (「天国だね」、「地獄行き」). 高天原 refers to the 神道 concept and is not really used in everyday language.
Kind of, but there's controversy and we ourselves don't know what it exactly is either. Since shinto myth is a mystified form of our ancestor's history, it's probably a place in north Kyushu. It's of course not a place above the sky where you go after death.