Is there a way to prevent a listing from being removed by the user? I mean, is there an option to allow only admins to remove listings?

@morfik
Well you can try to hide all remove buttons but still not supergood.

You could go to model/Item.php in core, find this line:
  public function deleteByPrimaryKey($id) {
and after it put:
if(!osc_is_admin_user_logged_in()) { return false; }

Hmm, I thought maybe there's some option I couldn't fine in the Osclass settings, similar to disabling a listing (allow user to disable their listings).

No no such ootion exists.
You may use history logger plugin if you need to keep historical data - what listings you had