HelpDesk 3 has many options like sending emails via PHP Mail(), SMTP or fetch email accounts to create tickets and so on.
Very important is that you do not mix or use the same email address for the same features.
When using PHP Mail() to send emails, please make sure that your main email address is registered on this server otherwise no emails can be sent.
When using SMTP to send emails, please make sure that your main email address is set under your settings otherwise no emails can be sent.
For each PHP IMAP account make sure you use an email address that is nowehre else used in the system, a complete isolated account only for that purpose. Do not connect this account to any other email client or programm or the emails won't be pulled into the system.