I've also been eyeing my custom fields with scheming eyes. So far, I haven't done more than write
a plugin to select which fields are shown in the file information box.
For the dropdown/checkboxes when using editpics.php, you'd have to modify the script or use a plugin to replace the current <input> with your desired <input> (using the "page_html" plugin filter since editpics.php doesn't have any plugin hooks directly in it). For modifying the script, you'll see that the form is generated using an array of fields and then functions to create each field. You'll see the custom fields in this array - for example here's the first custom user field:
array($CONFIG['user_field1_name'], 'user1', 0, 255),
There is a function to display the album dropdown box, but that function is particular to the album dropdown so you cannot re-use it for your own dropdown. You could use it as a model for writing your own function to create the dropdown or checkboxes, as desired. Make sure to put comments around all lines in your modification so you can easily apply these modifications to future versions of Coppermine.
For display of the fields (including hyperlinks), the plugin filter to use is "file_info". The plugin "Display Fields" (mentioned above) would be a good example to show how to do so since it allows you to unset file information fields. Change that to showing hyperlinks or whatever for particular fields and you'll all set.
You should also be able to do the display with a theme customization, but it might be difficult - I'm not sure without looking into it more. I would go the plugin route because that would be fairly easy.
If you want to write the code, go ahead. Otherwise, I cannot say when I'll get to it. It's on my "to do" list, which is unfortunately quite long.