Would it be possible for the user to set his listing to inactive?
A logged in user can go the listing page to see all the items he has added.
But I can't see a link to set the status of the listing to inactive, only a button to remove the listing.

In osclass 4.x, you can enable users to deactivate listing.
This require theme modification and guide can be found in sigma theme

Nice I got the button added to the items page!

Would it also be possible to add a different status to the ads?
Because now you have Inactive, Active and Expired.

Only when you add a new ad and need to check it before it's posted on the website you see the add as inactive (and no text that it's need to be checked before publication), and when you mark a as inactive it also has the mark inactive.

Maybe it would be better when a new ad is posted and needs to be checked by the admin the status of that ad would be pending instead of inactive?

You can use custom fields for that.