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The readings of "shi" for 4 and "shichi" for 7 derived from Chinese. Such readings are called "on-yomi" (sound-reading), because they are read in accordance to the sound of the Chinese language. They might be different from today's Mandarin, but it is said that such reading existed in the Central China in about 3 D.C..
"Yon" for 4 and "nana" for 7 derived from essential Japanese, which has existed before Japan introduced the Chinese culture. Such readings are called "kun-yomi" (orthodox-reading). When the kanji was imported from China, we allocated this Chinese character to the corresponding Japanese word and read it in correspondence to the Japanese sound. That is the origin of kun-yomi.
As you have mentioned, "shi" and "shichi" are the main readings. While "yon" or "nana" is used when the rhythm in reading the sentence needs such reading, or when accurate communication without mistaken listening is needed.
We hear such readings in stations and so on. For example:
●Nana ji yon-juu-nana fun hatsu Nagoya iki wa yon-ban sen desu.
(The 7:47 train bound for Nagoya is on track No.4.)