You wrote - My icloud storage is almost full, so I am trying to download my photos to my iMac (which is too old to automatically connect to icloud). When I go to the icloud web application it will not open Photos.
See this article iCloud: Delete photos and videos to save storage
iCloud: Access your photos and videos on iCloud.com
I first get a message saying, "Network unavailable or slow. - read this article How to troubleshoot Wi-Fi connectivity - Apple Support
If you are viewing this screen shot -
It happens sometimes when there is a glitch on the service provider network , open system preferences and click on network > advanced > TCP/IP click on renew DHCP see this article Internet connection does not work with an IP address space of 169.254.xxx.xxx from ISP, router, or manual configuration …
A question the issue you are facing is it permanent pop up or occurs sometimes for a very short while .
Also incorrect dns server settings , some proxies , vpn services are set up could cause this issue .
You must also create a new user account and just test , is that problem occurring in this account see this article How to test an issue in another user account on your Mac - Apple Support
In the last don't ever use any anti virus , any type of cleaners , softwares , not a single third party applications in your Mac .