No, both your preferences are currently not supported by Spiceworks Cloud Help Desk (CHD). There exist a few feature requests around your preferences. The most prominent one is this one, relating to your first point. Feel free to vote up and comment if you feel this request appropriate for your use case or your preferences. Spiceworks uses ranking of feature requests AND comments for determining priority of implementation.
For your second point, there exists an organizational option to come close to what you want. As Spiceworks techs will only see tickets assigned to them, you'll need to ensure that a dispatcher assigns tickets to Spiceworks techs according to organization. The dispatcher has to be a Spiceworks admin and will see all tickets though. And the Spiceworks techs will not see any unassigned tickets neither, even if intended for their organization.
If you need such strict separation already now, you've the option to create a CHD instance per organization. But you must then provide as many email addresses to your staff as they need to access CHD instances. An email address may only join one CHD instance. There exists a feature request to remove this limitation (although it seems that you like this limitation). There does not exist a limitation on the number of email addresses a person may have.