Can I save more fields to a Template beside summary and description
Yes! Try it live: you can save Reporter, Components and more!
How can I create template for Epics with multiple tasks (not sub-tasks) in it?
There's no such feature yet, but we are working on the ability to create issues from multiple selected templates, so that it will cover that case.
Why I can't find "Create from Template" or "Manage Templates" menu items?
There's different implementation for such functionalities for Cloud and Server instances. Make you sure you spot at dedicated section of Create Issue from Template and Manage Tempates articles of our knowledge base.
Manage templates for Server requires enabling "Bulk Change" Global Permissions for "jira-software-users" group. Check it out:
How can I help a team to better troubleshoot an error