- It is NOT necessary to have an agent on the Core.
- It is also NOT a problem to have an agent on the Core. No need to panic.
- Yes, you CAN uninstall it by running "UninstallWinClient.exe" without a problem. Note that the binary WILL reboot you, so be prepared for that.
... all depends on whether you want/need to manage the Core itself & make sure that you've got the right components with the right settings installed (as you'd probably want to treat it more like a server than a desktop client).
... it's not a problem . There were issues (some 10-15 years ago) with that, but not for many years & versions now.