If you hover over the crate name you can copy the URL of that too. We've made the URLs pretty hard to guess but will add further protection later (and welcome tooltips to make sure people know about stuff like this).
It took me couple of minutes to figure out how to copy the link to the URL. You can copy the url automatically to clipboard right after user uploads the files (and display the notice to communicate it) or provide a visual clue to make it easier to find URL copy feature.
Oh, smart! Never thought of that. We'll look into a nice way of implementing that.
Be careful with this. A lot of times I have something in the clipboard that absolutely can't be overwritten. Automatically writing to the clipboard isn't exactly expected behavior.