You say the email template is contained in a public folder. I seriously doubt the force.com site users have access to this folder. But presumably the trigger should run in the context of the case owner, which I'd hope is not the force.com site user. Are you creating any kind of assignment rule?
I strongly suspect what's at the root of your issue here is that you're not using the declarative functionalities built into Service Cloud for case creation and assignment for cases created from your force.com site. I see you were using the Community Tag in your post which I've edited and change to force.com sites. In either case, you really shouldn't need to be inserting these using a trigger at all. Is there a reason you're not using Web to Case? That would automatically insert these cases for you from either a site or community.
You don't need to give email permissions to your user to send them an auto-response email. They're not the ones who are sending the email, the system is sending it to them. You've also tagged your post for Salesforce1-app. Is there a particular reason for that? What does it have to do with your question? I recommend you change your site to use the web to case form either within a component if you wish or embedded in a standard visualforce page that sites uses. You'll find the documentation here.