All of my pages have class "page--node-" on the body element. This doesn't seem right
Classes beginning with page generally come from drupal core in template_preprocess_html(), which calls theme_get_suggestions() to do the heavy lifting.

That function makes a bunch of suggestions for the classes or template suggestions based on the path of the current page. The % is used as a wildcard for numeric arguments (in this case the node id).

Then back in template_preprocess_html() it passes the suggestions through drupal_html_class(), which in turn runs them though drupal_clean_css_identifier(), which strips out characters not allowed in a class, including the %, so you get a class of page-node- (this seems like a bug but there is a use for it I will explain below).

This theme_get_suggestions() code is the same reason you have the template suggestion with the percent symbol.

This means you can affect all node pages by using the template name page--node--%.tpl.php.

That might not seem necessary because you also have page--node.tpl.php, however if you want a different template for the path /node than you do for individual node pages you need to use both page--node.tpl.php and page--node--%.tpl.php.

This same reason is why the unusual class is useful.
If you want to style the /node page differently from individual node pages you can style page--node differently from page--node-

I agree it is a silly looking class though with pretty limited (or maybe none) documentation.

Also, something I found after writing this that is useful is How Drupal determines page theme hook suggestions based on path, from the "Drupal 7 Template (Theme Hook) Suggestions" documentation page.