[SalesForce] ‘INSUFFICIENT_ACCESS: portal user email settings are not available’ error on System.setPassword

As you pointed out, the Receiving_User is actually a Contact object - this means there is no Username field on that Object. Depending on how you set up your users, I would assume that the Email field is actually used for generating usernames for customers. In this case, you can just change {!Receiving_User.Username} to {!Receiving_User.Email} and this would be merged with the user's username.

If that is not working, what you might do is use a visualforce template with an apex:component, supply the {!Receiving_User.Id} to the component's controller, query for the User with that ContactId and then return this User.Username to the VF-Email.

Yes you do. See Customize Support Settings. In Setup, you'll need to go to Customize > Cases > Support Settings. Click on edit and check the boxes labeled

• Case Creation Template (select a template if not already checked)
• Send Case Notifications from System Address (impacts a number of things, see docs)
• Enable Default Email Templates or the Default Handler for Email Action.

Note: You'll need to use the look-up dialogue to select the email handler class you'll be using to send your emails. You can also use that checkbox to select the merge fields in certain cases. Your community users would presumably need sharing access to that class and the templates if you want them to be able to generate those emails when cases are created from the community. I suspect that sharing permissions is the likely cause of your issue.