No, you do not need to do that. All steps will proceed completely online. The Government of Oman approves up to 161 countries for all kind of Omani eVisa. Please check requirements for details.
No, you are not permitted to apply in that case.
The Omani eVisa is not available to Diplomatic/Official Passport Holders. There is just only the ordinary passport is able to use for apply Omani eVisa.