As for the issue of names, it really involves two questions - what to call the book, and how to spell it?
I Ching or Yijing? 国际易经网iqing.netI Ching and Yi Jing are just alternative English spellings of the same two Chinese characters, meaning 'Change Classic': the Classic of Changes. There are two methods of representing Chinese characters in English. The Wade-Giles method is older and more commonly used, giving familiar spellings like 'I Ching' and 'Tao Te Ching'. In the Pin Yin method, these are written as Yijing and Dao De Jing. Neither method is the 'right way', but in practice the Pin yin method tends to be used to imply a more modern, scholarly approach. Incidentally, the title can be written as one word or two, though Yijing is more common than Yi Jing, and I Ching is more common than Iching.
I Ching or Chou I? 国际易经网iqing.netChou I and Zhouyi are the Wade-Giles and Pin yin spellings respectively for 'The Changes of Chou'. This is the name for the earliest parts of the oracle, as it was developed and consulted during the Chou Dynasty (circa 1100-400BC). Later, during the Han dynasty (206BC-220AD), the oracle was given the status of 'Ching', a Classic, canonical work, and a wealth of commentary was added, completing the Book of Changes as we have it today.
While the complete book is occasionally referred to as the Zhouyi, this name is more normally used to identify the earliest strata of the book as opposed to later additions, often by people who are interested in recovering original divinatory meanings that were unknown to later commentators. The name 'Chou I' is actually not often seen, with most scholars preferring the Pin yin version, Zhouyi.