As I guess you know, "Spartan" was developed in parallel to Internet Explorer 11. On April 29, 2015, during the Build Conference keynote, it was announced that "Spartan" would officially be known as Microsoft Edge.
Internet Explorer was the default web browser for the earlier versions of Windows 10, but after the Windows 10 Anniversary Update on August 2, 2016, Microsoft announced that Internet Explorer would be deprecated on Windows 10 and "Microsoft Edge" was decided to be the main default web browser.
The main difference between Internet Explorer and Microsoft Edge is the engince they are using. Internet Explorer is using the legacy Trident engine, while Edge is using EdgeHTML as engine. It's basically a fork from Trident that has removed all legacy code of older versions of Internet Explorer and rewritten the majority of its source code to support web standards and interoperability with other modern browsers.
Complete history here: Wikipedia
Anyway, they are different but they are sharing some of operating system resources as you'll see on my next answers.
Answering to your questions:
- Does IE and Spartan share their cookies from each other, or are they separate?
No, they are separate.
- Does Microsoft Edge authenticate all subsequent Winsock connections once the first connection is made, like IE does?
Yes, it works in the same way.
- Are root certificates shared between Edge and IE? Or are they separate like Firefox?
Yes, they are shared because they use the certificate store managed by operating system.
- Are any passwords shared between Edge and IE? (Including Enterprise credentials, or Microsoft Accounts/Live.ID/Passport)
Yes, they are shared.
- Are the file caches separate in independent from IE and Edge (possibly leaking privacy information with cached files)?
Yes, they use same cache files.
- Are IE and Edge proxy settings shared?
Yes because both are using the operating system's proxy configuration.
You can configure almost all of these options (certificates, proxy settings, cache and temporary files settings) from the same place. Press Win+R, type
inetcpl.cpland press Enter. Certificates are under "Content" tab. Proxy settings are under "Connections" tab, "LAN configuration", and Cache and Temp files can be managed under "General" tab pressing on "Configuration" button. I'll put you some screenshots. Sorry, they are in spanish like my system, but I put some tags in english: