1) Give them the permissions to do so via groups unders Admin Panel > Agents > Groups and assign the group permissions then to their agent accounts
2)
Enable ticket auto-response for user and agents (student, teacher and mentor in your case) and auto assign tickets to a team consisting of agents (teacher and agent)
Take a look at:
Admin Panel > Settings > Autoresponder
Admin Panel > Settings > Alerts & Notices
Admin Panel > Agents > Teams
3) Yes, just make their accounts to Agents (and not admins) under Admin Panel > Agents > Agents > Name of the Agent, e.g. John Doe > Account Type > Agent