would be nice if I could access the IL page of the level of the run I'm currently reviewing, so I don't need to: open a new tab go the page of my game go the IL page click the IL I want to check.

Very tedious.

right click the level name -> open link in new tab or just press middle mouse button

this is assuming you are using: speedrun.com/runsawaitingverification

if you're using moderation hub you will need to click the view button first

You can click View run then from the page opened click the category name - if I understand what you mean correctly