Can Listings Be Auto Approved After Submitting ?

Yes, we have the option available for it.

Please go to Theme Options > Listings > Submitted Listing Require Moderation

Set it as No.

When a Free listing will be submitted the user will have a publish button there on the preview of the listing where he can directly publish the listings.

In the case of paid listing after payment is made and it goes successful listing will be automatically get published.