I read that, starting in Word 2007, VB6 is not functioning anymore.
Wherever you read that, it's wrong. Word 2003 and Word 2006 both run
VBA6 (not VB6; "Visual Basic for Applications" and "Visual Basic" are different
but related languages). Word 2010 has VBA7, which will run everything
written in VBA6 and more.
My question is: when I migrate from Word 2003 to Word 2007 would I
have to rewrite my VB6 macros in another language?
No, you wouldn't have to rewrite them in another language (and I can't imagine
what language that would be -- Urdu, maybe?).
But there are some things that may have to be fixed up. For example,
Word 2007's new file formats use different extensions (.dotx, .dotm, .docx, .docm).
You could just stay with the old .dot and .document formats -- Word 2007 will
read and write them -- but you wouldn't be able to use any of the new features
the new formats support. If any of your macros modify menus or toolbars, they
won't work with the ribbon in Word 2007. There are a few other gotchas, but
for the most part the old macros will work just as before.
My second question is: can I use Word 2003, including my VB6 macros,
in Windows 7?
Yes, absolutely. I have Word 2003, 2007, and 2010 running side by side
(and sometimes simultaneously) in Windows 7 on several computers.