I think Han and Shu comes from the chinses traditional solar terms. The hotest days in a year are called Da Shu which contained in summer vacation and the coldest days are called Da Han which are in winter vacation. However, both Han and Shu are old words, we dont use them in the daily life, just say Re and Leng.

The reason is similar to English words, "holiday" vs "vacation". Of course there are technical differences in nuances but no difference in general usage. As for your 2nd question, you used "get away". Technically different but we get the same general meaning. 夏/冬假 are perfectly acceptable. You can say, 夏天拿假. 聖誕息期 can mean that companies shut down for two weeks during Christmas time. In this sense ,the workers get a mandatory vacation! Best to stick with the cultural preferences of the language instead of trying to use some other ways of expressing unless you are writing a creative piece.

For the second situation, you can just say that you will go on a Xiu Jia, which means "Rest Vacation" in English.

1.寒假和暑假是针对学生来说的，只有学生有寒假暑假，春天和秋天都在上课，不放假。 2.我们广大劳动人民是不放假的，如果春天或秋天放假，就说“春天放假”，“秋天放假“。你也可以说”春假，秋假“，但我们不这么说。 3. 寒假，暑假 是固定说法，不说夏假冬假，就像have breakfast,为什么不说eat breakfast呢？

中国正式出现寒暑假制度是1901年，当地人们使用的是阴历和二十四节季计算时间，因此call为寒、暑假。