I deployed the control list with paging and item template for using the with Content Search Web Parts, we have realized that users with no Full Control are not able to run the logic in these templates. SP admin has told me that this is not correct because the users won't be configured with Full Control, so my question is, what else should be configured in order to get the scripts running? My understanding is that all the users that have access to the folder will be able to run the scripts?
Yes, in most cases they should be able to, unless running scripts has been disabled by an admin or they do not have access to the place where you placed the script file. Because the file has been published, right?