First off you havne't given us very much to go on as far as specifics about your setup. (Like server OS, etc)
1) I don't know why, but we do not receive emails from the customers...
Have you configured OST with the email address credentials and server?
Have you configured cron or windows to run cron.php so that it will actually try to collect mail?
2) Whan we try to send from Osticket an email, then we see that Systems took a name of our HOSTING, not our text in the topic of email.
How do you have OST configured for sending mail? (SMTP or PHPMail)
If its PHPMail under Windows have you configured PHP for it? Look in your php.ini for "SMTP ="
