Having 2 enclosure per RSS <item> is not specified in the RSS specification. But it is at least for iTunes no problem - as far as I know it. iTunes takes in such cases the first enclosure of an item and podPress rearranges the enclosure so that the podPress attachment is the first one.
Another part of the issue is that WP creates automatically enclosure when you insert links to media files into the post content. It does that when you publish the post. WP creates a so called Custom Field with the name enclosure for each linked media file.
But you can delete the custom field and update the post in order to have only the podPress attachment in the feed.
The download text link and the icon image both lead users to a blank white page that later loads a player
That is the default behaviour if you set a link to a media file.
but doesn't allow them to actually download the file.
That is incoorect. When the browser shows the player then one can click in the main browser menu bar (or buttton) and choose the option "Save page as ..." or "File > Save as" to save the media file.
On devices with a right mouse click functionality one can make a Right-Mouse-Click on one of the links and choose "Save as ...". Something comparable is possible on Android devices. One can touch on one of the links and hold the finger on the link, a menu appears which offers you also to save the file.
To force the download on every click isn't that easy and I'm already thinking about a good solution. I will maybe implement something. But I can't say when this will happen.