Should mention that it happens whether or not I'm interacting with Firefox, too. I can be completely idle, doing something on my phone and I'll see it pop up or it can happen while I'm typing in this box right now. There's no rhyme or reason for it.

Set this setting to false in about:config

browser.download.improvements_to_download_panel

It seemed to fix the problem for me and for many others.

WARNING: Changing preferences through this interface not officially supported Hidden settings edited using the about:config tool are explicitly not supported, which means that Mozilla makes no guarantees they will be supported in the future, or that Mozilla will fix them if they break. Mozilla does not test these preferences, and will not in the future. That includes security and performance testing which these preferences may affect.

[Warning added by moderator]

That change will be temporary (sorry, don't have the link for that). Set the following preference to false. browser.download.alwaysOpenPanel

https://support.mozilla.org/en-US/kb/about-config-editor-firefox

WARNING: Changing preferences through this interface not officially supported Hidden settings edited using the about:config tool are explicitly not supported, which means that Mozilla makes no guarantees they will be supported in the future, or that Mozilla will fix them if they break. Mozilla does not test these preferences, and will not in the future. That includes security and performance testing which these preferences may affect.

[Warning added by moderator]

I have been using the first solution all day and it's worked fine. But after doing some searching on both of those options it seems like the second one is really the cause of the problems that I was having and I am far from the only one. My search terms were just not hitting on the right posts.

Opening at random like this must be a bug, though.

Thanks both of you.