Almost every Chrome action opens a dialog box that says "Allow opening file? Allow snap "chromium" to open file "/home/iii/Downloads"?"

I've run into the same issue, and in my research on the issue, I've found no solutions (outside of just not using the snap version of Chromium, which doesn't even work on Ubuntu 19.10, as installing chromium-browser via apt will install the snap), but plenty of questions asking for one.

Your title (emphasis mine),

Almost every Chrome action opens a dialog box that says “Allow opening file? Allow snap ”chromium“ to open file ”/home/iii/Downloads“?”

refers to both Chrome, and Chromium. I'm guessing since you're having this issue, that you're using Chromium, not Chrome, as I don't believe there is a snap for Chrome.

According to this answer, this is "issue", as we see it, is actually an intended, hardcoded feature that cannot be changed without changing the source code of snapd. (Emphasis is added.)

This is the intentional behavior of snapd and hard-coded into source code.

This feature, opening files in snapd, is new (about seven month old). As I understand the developers are conservative and [the] prompt is to prevent any security issues. The discussion about subject in forum.snapcraft.io and pull request that introduced new feature.

From the above-mentioned discussion on the snapcraft forum (again, emphasis added):

The /usr/bin/xdg-open in the core snap will now accept local file paths, including relative paths. It then calls the new io.snapcraft.Launcher.OpenFile D-Bus method, passing a file descriptor to prove that the confined application has access to the file. Outside the sandbox, snap userd will show a graphical permission prompt and then launch the file using the real xdg-open.

This feature also supports opening directories, causing them to be opened in the file manager.

If you look at the relevant portion of the source code of snapd, you can see that there's no way to bypass the above graphical permission prompt, without adding an additional check against a global or app variable that you set in order to automatically allow it. As it is, it's impossible.

This all applies to Chromium (or any app) installed by snapd, and due to the changes, as well as Chromium installed via apt in 19.10. However, for Chrome, you can use the .deb installer you get from simply downloading Chrome from google.com/chrome. Since it's a self-updating program (I believe so, I'm not 100% sure on that), you needn't worry about it not having a way to update via snapd or apt.

So, if you really, really hate the prompt, you can get around it by installing Chrome, rather than Chromium. There are some differences between Chrome and Chromium, some good, some not so good. It's mostly the same, though.

To be clear, this isn't a solution. This is stating that there currently isn't a solution, as the "issue" is considered a "feature" for security purposes by the developers. If you're on 19.10, and can't stand it, your only "option" is to move to Chrome or another browser that isn't installed via snap.