How to find out the URL of my search results and edit the columns in the results?
You can configure profile to use as contact search. You can also control the result columns in profile search. To display contact result by default you can append force=1 in url. 


Ref: https://docs.civicrm.org/user/en/latest/the-user-interface/searching/#views-for-display-contacts


You can create smart group which holds all the contacts and use the contacts link from Manage group page to use as a url.

Use contact search url and apend force=1 to display contact by default.


HTH

Pradeep