You have listed a large number of issues, many of which point to something being fundamentally broken with your site.
However, the photonic plugin was having problems. It no longer showed the pictures on a page in edit mode, although it would display them when viewing the page.
This typically happens when you have a plugin conflict in the back-end. Note that Photonic doesn’t show pictures on a page in the edit mode anytime – what you see is just a placeholder logo (i.e., for Flickr galleries you will see a giant Flickr logo in the edit page). The only place where you see photos in the back-end is in the “Gallery builder wizard”, which lets you pick what you want to show.
Also the galleries were throwing error about a CurL call not working.
If cURL is not working on your site the folks to help out would be your hosting people. Note that very only one thing in Photonic actually directly requires cURL: the Google Photos module for cases where you are explicitly defining search filters. For everything else cURL is called by WordPress internally. Depending on what error you were getting, the fix would either be at your host’s side to enable cURL, or to increase the cURL timeout (Photonic lets you do that, and you might have been facing a timeout because of your shared hosting).
I enter them on the settings page, scroll to the bottom and click the save page button, it redirects me to the How To page, and when I go back to the Flickr settings page both fields are empty.
If you save a settings page in Photonic, you are brought back to the same page, not the “How To” page (which is the first page). This leads me to think that you have some sort of security plugin that is stripping out query parameters when you are trying to save. This could, in some cases, also be due to some settings at your host’s end, though I can’t think of anything unique – Photonic uses standard WP APIs to save settings, and in the 9 years of the plugin I have very, very rarely come across an issue where users were unable to save settings.
You are right. It looks like I was confusing the standard Flickr image plugin control with your gallery control. The Flickr image plugin does not show previews in edit mode, which is odd and annoying.
Your plugin seems to be broken because my host has curl turned off. Thanks for directing me to what the issue was.