You shouldn't have a problem adding users with different domains to a single organization.
But I'm not sure what you mean by wanting to add multiple sites to an Organization. Do you mean when you create New Organization that you want the website field to have more then one domain? If you do then you have control over the Website field by going to Admin panel -> Manage -> Forms -> Organization Information.
You can change the Type to Long Answer.
You cannot have opening a ticket update this field however. It's organization information and can be edited by people with privs to update it.
You can always add a field to Ticket Details for the site(s), that Agents can fill in when a ticket is opened, or editable on existing tickets.