The following are the privileges of a project user
The following are the privileges of a client user
Note: Client users cannot access Timesheets
The message user already added as client to this organization is displayed because you would have deleted (removed) the user only from the project and not from the entire organization.
To delete a client user :

Yes. You can delete a project / client user and can add them again as a client / project user. For this follow the steps below
To add a client user :
To add project user :
The invite mail is a system generated mail sent from Zoho Projects along with username and password. Sometimes this mail would have landed in your SPAM folder. Hence we request you to check your SPAM folder for the invite mail.
If you are not able to trace the mail kindly mail to support[at]zohoprojects[dot]com for further assistance.
This is how your invite mail sent from Zoho Projects would look like :

The portal owner who invited you to their portal would have forgot to add you as part of their project hence access is denied for you. To get an access, you need to request your portal owner to add you as a member in their project.
Only milestones marked as External in Milestone flag is accessible for client users and not otherwise.

No. Currently, the project can be assigned to a client company only. i.e Any of the client company added to the project can be made as the Primary Client.
No. The user roles and its access privileges are predefined. As per the existing design you cannot modify or define new user roles.
Client Company and associated users can be added by the Portal Owner, Administrator and Manager. If you are in either of the aforementioned roles, you will be able to add the client company and users.
Portal Owner and Admin :
Manager :

It is not possible to pull name from your active directory to assign users. The users must be added into your projects portal in order to have them assigned to the tasks and other project activities.