How should I convert Markdown (or similar) to a Word Document?

Get pandoc version 1.9 or higher. Then:

pandoc myfile.txt -o myfile.docx

Recent versions of pandoc allow direct conversion of markdown to word.

This isn't a complete answer to your situation, but you may want to investigate using Pandoc which can convert Markdown into other formats including RTF (which MS Word should be able to handle reasonably well).

Unfortunately, I have no idea if this will work with the DeepFreeze system you describe.

If you already have a Markdown tool for producing HTML, open the HTML in a browser and copy/paste into Word. Alternately, just open the HTML file itself in Word then save as Word format.

The method that has worked best for me is to use markdown to produce HTML, upload the document to Google Docs, select the option for Google Docs to convert it to their format, then download it as a DOC file.

Using pandoc to convert to RTF works (make sure to use --standalone option), but it loses the "style* of headers in Word.

Using Google Docs, the resulting DOC file keeps the notion of style so that you can change the formatting of all h1 headers in one step.