Hi SeidKrv,
Thanks for your update.
Following article introuduces the permissions that need to assigned before running "New-Mailbox" command.
Please focus on "Recipient Provisioning Permissions" session.
Recipients Permissions
http://technet.microsoft.com/en-us/library/dd638132(v=exchg.150).aspx
Based on the article, it seems both Recipient Management role and Organization Management role are required.
More detailed information on both management role as below:
1. Administrators who are members of the Recipient Management role group have administrative access to
create or modify Exchange 2013 recipients within the Exchange 2013 organization.
2. Administrators that are members of the Organization Management role group have administrative access to the entire Exchange 2013 organization and
can perform almost any task against any Exchange 2013 object, with some exceptions. By default, members of this role group can't perform mailbox searches and management of unscoped top-level management
roles.
Thanks