I am curious to know how bilingual people use multiple languages on their Linux computer. Does anyone know of a good book or paper explaining how Linux functions to allow users to use more than one language?

I am looking to answers to these questions to satisfy my curiosity: How can a person type chinese characters with a QWERTY? How are applications tranlated to multiple languages? Is it recompiled? Why do I end up with garbage in my Web browser unless I change the character encoding? How can right-left and left-right languages be mixed in the same word document? Are there tools that Linux lacks or performs poorly in regards to these issues?

I can only speak for Japanese, but most if not all Linux distros seem to handle it well. Displaying foreign language characters has never been a big deal AFAIK - all you need to have is the appropriate font sets and character encoding. As for typing Japanese characters, you can use the standard QWERTY keyboard and have the sequence of Roman alphabets converted to the appropriate characters via one of the conversion engines; i.e. Canna, Anthy, etc.

Gettext, usually. You don't have to recompile to use different versions. All messages are kept in separate files. The right file (with the right translation) is choosen using environmental variables, but usually can be re-loaded at request (the program has to support that option, through).
Default encoding or fonts problem. I'd make UTF-8 default. That should help.
In Word document - I'm not sure. In Linux, when the document is in UTF-8, I see no reason why not. The program used to edit the file must be able to switch, however.
Support for less common languages (not counting users, but developers) is not always as good as it should. Especially when we think about Asian scripts. It'll come, I think, with time.

It's not always the case with Japanese Web sites. Many of them still use Shift-JIS (thank Microsoft for THAT), with some others using EUC-JP and ISO-2022-JP. Choosing "automatic" for your browser encoding usually works, but again, not always.