>> In one column would be a small calendar grid for the current month (with arrows for upcoming and previous months). Dates with an event attached would appear in a different color, and details would pop up in a ballon on mouseover, or possibly below upon clicking (it would be nice if users could easily paste/print this information).
Have you considered using a Google Calendar? It does similar to what you say you need. And it can be inserted into your own website.
>> In the other column would be links for the uploaded documents, with the most recently uploaded appearing at the top. However, instead of scrolling down on an endless page as we currently are doing (the users do not want me to expire old uploads off the site) there would be a keyword search field above the first page of links that would search all uploaded documents.
OK, maybe look at Backpack instead - http://www.backpackit.com/- get a free account to test it, it's a simple Intranet solution that has calendars, file uploads and a lot more besides.
>> Finally, I'd like the whole thing to have a more professional look than I'm able to create.
What's more important on a private Intranet? Ease of use or the visuals?
>> I've experimented with adapting a WordPress blog to this purpose, with each uploaded file as a post, but have run into some issues - can't upload Excel files;
You should be able to upload Excel files in WordPress, just like any other file - what goes wrong when you try?
>> the calendar widget tracks dates of uploads, rather than the events I want it to reflect.
Test drive a few of the many calendar plugins available for WordPress - maybe http://wordpress.org/extend/plugins/calendar/.
>> Also, this blog is on the WordPress site. I'd rather download the WordPress software (which our webhost does support)
it should be inexpensive to upgrade your hosting. If your hosting company doesn't support php and mysql then considering moving elsewhere. Most open source CMS will require these technologies.
>> and host the private blog on our own domain, but it's unclear the privacy functions are supported in the standalone version.
You can make pages private if WordPress is hosted on your own domain. If you need greater control over permissions there are plugins to enable this, or you can set additional passwords on specific folders using your domain control panel.
>> However, the subdirectory into which I would install it is already private, so perhaps that wouldn't be an issue? It seems to take a lot of research to find answers to apparently simple questions about configuring WordPress, which concerns me.
There is a huge amount of support and information available for WordPress, I can usually find answers to anything in the Codex or support forums. But sometimes you need to type in the right jargon to find it!
>> Any thoughts on whether WordPress is an appropriate tool for my goals, or something else would be better?
I think it can do what you need to do. There are many alternative CMS that are equally good. But on the whole I think Backpack might be the best bet - no web design required.