To create a PDF with Acrobat Distiller or Writer, is the original
authoring software required? For example, can you create a PDF file
from a Quark Express file without having Quark installed on your hard
drive?

The following answer is based on information found in the help files
that come with Adobe Acrobat 4.0.  I have checked Adobe's website for
any relevant changes in the latest version of Acrobat, but it does not
seem that the possibilities have been expanded.

"About PDFWriter and Distiller
Acrobat can use either PDFWriter or Distiller to convert your files to
PDF. Both utilities are included in the default Acrobat installation.
PDFWriter is a printer driver that converts files directly to PDF from
another software application.  Distiller is a tool that converts
PostScript files to PDF, and in many cases it provides higher quality
output than PDFWriter.  (In some installations, Distiller can also
convert the original file to PostScript and then convert the
PostScript to PDF.) PDFWriter is often quicker to use than Distiller.
[..]

Creating PDF files with PDFWriter
PDFWriter "prints" a document quickly in PDF. You can drag and drop to
create PDF files with PDFWriter or use the Print command in your
authoring application. [..]

To create a PDF file with Distiller by opening the file in Acrobat
(Windows):
Choose File > Open, choose All Files from the Files Of Type pop-up
menu, select the file you want to convert to PDF, and click Open.
You can open a document from Adobe FrameMaker, Corel WordPerfect, or
Microsoft Word, Excel, or PowerPoint; an HTML or text file; or an
image file of type GIF, JPEG, TIFF, PCX, PNG, or BMP."

To sum this up: PDFWriter behaves like a printer driver - if you can
print a file, you can convert it into PDF.  Distiller requires a
postscript (.ps) or encapsulated postscript (.eps) source file, or one
of its supported file types (as listed above).  You cannot generate a
PDF file from a Quark Express file (.qxd) without installing Quark (or
another application that can print Quark's files).

Adobe's website has some online documents describing how to get the
best results when generating PDF files from Quark:
http://www.adobe.com/support/techdocs/76c6.htm (Windows) and
http://www.adobe.com/support/techdocs/1cd72.htm (Macintosh).  (These
documents will only be useful if you are able to install Quark on the
computer in question.)

Adobe also has an online PDF generator (https://createpdf.adobe.com/)
that offers 5 free conversions.  It supports the following formats
(http://createpdf.adobe.com/cgi-feeder.pl/formats):

"Microsoft® Office:
Word (.doc), PowerPoint (.ppt), Excel (.xls)

Other Microsoft Formats
Publisher (.pub)

Adobe Formats
Illustrator® (.ai), InDesign™ (.indd), FrameMaker® (.fm), PageMaker®
(.pm, .pm6, .p65, .pmd, .pmt), Photoshop® (.psd)

Corel WordPerfect Office Formats
WordPerfect (.wpd)

Adobe PostScript® Formats
PostScript (.ps, .prn), Encapsulated PostScript (.eps)

Text Formats
Rich Text Format (.rtf), Text (.txt)

Image Formats
Windows bitmap (.bmp), GIF (.gif), JPEG (.jpg), PCX (.pcx), PNG
(.png), RLE (.rle), TIFF (.tif)

Web Capture
Web pages (.htm, .html, .shtml)

Paper Capture
Adobe PDF (.pdf), Bitmap (.bmp), JPEG (.jpg), PCX (.pcx), PNG (.png),
TIFF (.tif)"

I hope this answers your question.  If not, please ask for
clarification.

Search strategy:
I searched through the help files included with Acrobat 4.0 and
support information on Adobe's website.

What a fast reply!