Yes, of course. Shannon's work was on the transmission of information over discrete, noisy channels. It centred on the trade-off between bandwidth and signal-to-noise ratio, and, grossly over-simplifying it, in particular what the maximum transmission rates could be with and without the presence of noise induced error and how to construct suitable codes. This involves coding theory1 and probability theory, which is where some of things that have you confused come from.
It is also incorrect that you need 5 bits to encode letters. If we have a known frequency distribution of the letters, we can encode the more frequent ones with fewer bits, and less frequent ones with more, which will give much shorter messages that the simple 5-bits-for-everything coding. For example, the entropy given by the usual distribution of English is 4.219 bits per symbol, so we can encode common symbols like 'e' with perhaps 1 bit, and uncommon symbols like 'q' with 5 or 6 bits, an on average we use less than 5 bits per symbol. If we use more cunning coding schemes, for example ones that take into account digram and trigram frequencies, we can get this average down to a little over 2 bits per symbol by using short codes for common two and three symbol groups.
Also, notably, your title is backwards, Shannon's Entropy is a measure of information, not the other way around. It is not the only one possible, however Shannon argues that it is the most natural and intuitive.
The full extent of Shannon's (seminal) work on information theory is too large to encompass in this format, but there is plenty of further reading. The wikipedia entry on information theory is reasonably approachable, and the page on coding theory also adds something. For the real stuff of course, you can just read the paper in which Shannon invented information theory essentially out of whole cloth; "A Mathematical Theory of Communication"2.
- This is not not suggest that Shannon was applying coding theory, he was inventing new ideas that would become large parts of coding theory.
- Claude E. Shannon, "A Mathematical Theory of Communication", The Bell System Technical Journal, 27:379-423,623-656, 1948.