Straight to your questions:
You can filter the Tickets list by any field. By default, the Organisation field is not included in the Tickets list, but you can easily add it.
The second one is trickier. The Plumsail HelpDesk has a web widget tool that gives your clients a way to view and access their tickets on a separate page. However, it operates on an email login basis, meaning that a given client will have access to his or her tickets, not to all the tickets from the same organization. A way around this is to create a separate email for all the organization's tickets and use it as a second requester email in each ticket from a given organization. Once you've got or created or such an email, the rest can be easily automated with triggers: when a ticket is created, the trigger will check the organization of the requester and add a second requester to the ticket. This way each client will have the ability to login to the HelpDesk web widget under the organizational account and view all the tickets.
Please let me know if you need any more details on this!