天干、地支、八卦和六十四卦 Heavenly Stems, Earthly Branches, 8 Trigrams and 64 Hexagrams
Brief answer.

Yes. Some modern Chinese words are developed from these conceptions.


坤 in 八卦 represents the earth, female, Yin.

From these meanings, words like 乾坤 (the world. lit. Heaven and earth), 坤角 ( female characters in Beijing Opera, but not necessarily played by a woman) developed.

甲 is the first element in heavenly stems, and 子 is the first element in earthly branches. The heavenly stems and earthly branches were popular to record the year in ancient times.

we have a word 甲子 which means 60 years, because the cycling length of  combinations of heavenly stems and earthly branches is LCM(12, 10) =60.

Yes, all most Chinese know these characters, however, some (at least some people I know ) can not remember the order exactly. It would take them few seconds to think of, for example, what the next element after 未? Today, in Chinese calendar, we still use this system to record the year. For example, this year is the year of 丙申 ( 申 indicates it's the monkey year).

Yes. You can see them in some idioms (成语) .



寅吃卯粮 To consume resources prepared  for or available in the future. Not planned well. Not prepared well. Lit. In the year of tiger, consume food prepared for the year of rabbit.
you can also see them in list bullets in articles. In English, you use a, b, c, d …, while in Chinese, you use 甲乙丙丁
in organic chemistry, 甲乙丙丁 … are used to express the number of carbon atoms in the nomenclature of substance. We have 甲烷 CH4, 乙烷 C2H6, 丙烷 C3H8…