Can we trigger the Case Assignment Rules on the creation of the Case? We are trying to avoid having to edit the case and checking the checkbox "Assign using active assignment rules".
No need for a trigger. You can do this directly from the case layout page. Edit the case layout that you want to have this box checked by default, and click Layout Properties.
You will see a 'Case Assignment Checkbox' section with two checkboxes, the first to show on the edit page, and the second is to chekc it by default. Check the box that you want it checked by default.