It appears that the second issue appears only on my local machine. After selecting default for the User agent String it was no longer set to Internet Explorer 6 when opening new logging windows.
Once the logger was loading with the correct User Agent String the console error issue was no longer seen.

I can reproduce this last issue only if I manually select the IE6 user agent string. If the log page is loaded without intervention the "Default" option is selected (which does not trigger the issue).

Please try again the test with a freshly opened log page.

Tested with ICEfaces 4 trunk, EE-3.3.0 maintenance branch r46945, Tomcat 8, IE 11 with localhost added to the compatibility view settings.
The global logging page is indeed loaded in Document Mode Edge as per fix however, when using the Download log feature WHILE the dev console is open an error occurs. At this point you can no longer interact with the page. If the dev console is not open when using Download Log, no issue is seen and the log an be downloaded.
See screenshot console1.png

SCRIPT5007: Object expected
File: log.html.jsf, Line: 107, Column: 17

Also it was noticed that the User agent string is set to Internet Explorer 6, not sure if this has any importance. See screenshot console2.png

Run copyToClipboard() and downloadLog() functions only when localStorage functionality is available to avoid JS errors thrown in IE7.

Carmen, in order to test this issue you have to add the server's host name to the list in Compatibility View settings (just as Arran described above). This will ensure that pages loaded from the specified hosts will run in compatibility mode (IE7). After the global logging page is loaded you can use the Developer Tools to verify what mode was used to load the page and if any errors are triggered when interacting with it.
As hopefully you will see the global logging page is loaded in Document Mode=Edge which proves that the meta-tag does its job.

Changing the mode in Developer Tools will always override the setup established by the page markup.

Modified copy to clipboard and download log functionality to still work when the local storage entries used for global logging are not created.

Re-opened for JS errors reported by QA.

Added meta-tag that blocks IE10/11 from emulating older browsers.