Must the account "short name" be all in lowercase? Why?

Yes. Lowercase alpha and/or numbers. The underscore is permitted (_) but avoid other special characters which can be a problem.

The short name cannot be easily changed so chose it carefully. Be sure you use a strong password if you are concerned about security.

..." what are the reasons?"...

In this type of situation, I would just go with "because Apple said so".

Since you have already created accounts that have upper-case characters in them, you probably found that they generally work fine in day to day use so already know it isn't strictly a "requirement".

But considering all of the environments Macs might be used in, and all of the protocols they might potentially use to communicate with other computers, it is possible that Apple is being conservative in their recommendation to reduce potential compatibility issues. Or the recommendation might be there because they know of some feature or old code buried in the OS somewhere that requires. And of course, since Apple has put this recommendation as a requirement in writing in a support document, some less careful developers may take this to heart and not bother to code for that possibility.

Although I never saw it documented anywhere, the short name "Administrator" (with just the first letter capitalized) was a reserved name in an earlier OS X version (10.3 or 10.4; I forget which - but it is no longer true in 10.5). So I suppose another possibility is that using upper-case characters in names is fine, but Apple has just reserved upper-case names for their own purposes.

Either way, Apple said to use only lowercase, so doing otherwise should be considered "unsupported" / "use at your own risk".