✔ E-Z Audit does not need to be installed to a server for any functional or licensing reason.
If you need multiple users to have access to the Admin Console, you can install it at as many PCs as you need within your company.
Licensing is based on number of PCs and/or servers you will be auditing.
The audit modules will get copied to a share on your server from your PC. See the E-Z Audit Users Guide for details.
✔ Audits do not launch from a console so your where you install to isn't relevant.
There's no "management" of audits that is required beyond a point and click, set it and forget configuration that takes seconds. The audit modules do not, and don't need to "talk" to the Admin Console.
✔ As a matter of best practices nothing that doesn't need to be on a server should go on a server.
We don't break best practices. Even software that has a reliability of 99.999% of not causing problems that may require a reboot or hard-shut down still aren't 100%. When that happens on your PC, you're not happy. When it happens on a server, a lot of people will be unhappy.
✔ Graphics issues will degrade the user experience.
In some cases it may not even be able to launch on a server due to screen resolutions being set to bare minimums. It will almost certainly look pretty bad - even remoting in.
✔ IE lock-downs that may disable some (or all) functionality for the Admin Console.
This is a bit old but still exists IE can be set to block apps OOB. If you see this then you'd have to create allow rules in Control Panel > Internet Options to allow the Admin Console (ezaudit.exe) to work.
This is NOT reporting a security risk, it's just a default setting to "block-unless-allowed".