A: Yes! A free download from Microsoft supports Windows 7, Vista and XP SP3, and despite its name does not require you to own Microsoft Office! 

In Windows XP you must enable East Asian languages. In Windows 7 and Vista, East Asian languages are enabled by default.
Install this update after Language Packs / MUI Packs for Windows or Office to avoid replacing this update with older IME files.
Go to this Microsoft Taiwan download page:
http://www.microsoft.com/downloads/details.aspx?FamilyID=60984ecd-9575-411a-bd38-2294f17c4131&displaylang=zh-tw
Although that page is titled "Microsoft Office 輸入法 2010", don't worry if you have an earlier version of Office or don't have Office at all. You'll still be able to install this (but see my notes about 64-bit Office below).
There are no important instructions on that page. Just click the button that says "下戴" ("Download"):

You will be taken to a second page. If the download does not start within 30 seconds, on the third line click the tiny little link that says "開始下戴":

The download page suggests an add-on called Intelligent Input Mode (智慧型輸入模式), which automatically switches between English and New Phonetic keyboards. This is not required and I haven't tried it yet. It's potentially very annoying; if you try it please let me know what you think:
http://www.microsoft.com/downloads/details.aspx?FamilyID=9DD0DB3A-FF78-4EBE-985B-8668EE93328F&displayLang=zh-tw 
After you download the file, double-click, install and you're done...unless you have trouble because of 64-bit Office.
I've received reports that Microsoft fixed the installation problems I experienced with 64-bit Office. If you don't see the following error message, you can skip this section and scroll on down.
When I clicked the installer, because I had 64-bit Office installed I got a very cryptic "You don't need this" error message. ---->
I've heard Microsoft has fixed this, but I haven't tested it yet so here's how to install the update with 64-bit Office (and not switch to 32-bit Office) if you see this same error message.
Option 1: "Painful Extraction"
I contacted Microsoft Support about this. They recommended I extract the 64-bit installation files, and install them using the method described on this Microsoft forum:
http://social.technet.microsoft.com/Forums/en-US/officeappcompat/thread/5e7c949f-b3e0-436c-a3ce-562fe9d1dfa6/ 
Option 2: "Painful Insertion (of credit card number)"
I didn't need to experience the painful option above, because I own the USD $24.95 Traditional Chinese Office Language Pack 2010
which contains the same Office 輸入法 2010 IME update, along with Chinese Proofing Tools and optional Chinese interface elements for Office. (The 2010 Language Pack won't work with Office 2007 or earlier, sorry.)
But, there was still one final annoyance. When I purchased the Office 2010 Language Pack from Microsoft's website, the download contained only the 32-bit pack! The installer tried to tell me to uninstall Office and reinstall the 32-bit version. A quick telephone call to Microsoft Store support (Digital River)
remedied this problem however. They gave me the 64-bit version, and all is well.