Is there an official D&D 3E character generator online or downloadable? (Not just a character sheet or tracker.)
If there are no official generators, is there a good one anywhere?
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There are three big ones.
Hero Lab from Lone Wolf Development is IMO the best character generator for D&D 3.5e and Pathfinder. It's being actively developed, has a large community, and is professional quality. It supports a wide variety of rulesets including 3.5e, 4e, Pathfinder, Mutants & Masterminds, World of Darkness, Call of Cthulhu, and Savage Worlds. You can download it and use it in demo mode, which doesn't allow you to save or print.
I have also used RPGXplorer and PCGen. PCGen is free and is decent, it's a freeware distant second to Hero Lab, and I really didn't like RPGXplorer, the interface was very unintuitive to me. You can try it yourself, it has a demo mode that works fully featured up to level 5.
The venerable (and free!) PCGen will make characters for multiple versions of D&D and similar d20-based systems.