Is it possible that the person I write to will see the email's content before I reply to him? If I got an email and I answering that email, is there any feature that can cause my reply to be seen by the sender before I send him back the email?
It depends on the email system. In general, no, this is not possible. Traditional email messages were not written on interactive websites (unlike, say, Facebook messages); there is also no central server you are expected to continuously be connected to (it's a decentralized system). So, traditionally, messages were written entirely client-side, in a program running on your computer, and nothing was sent over the network until your computer, at your command, sent the message.
Nowadays, it would often be possible as a technical matter for email to be read while being written; there has been a lot of email provider consolidation, and (for instance) Gmail can keep track of what you're doing on their site. Google could, if they wanted, record if you're typing (or even what you're typing) and tell another person connected to their servers. However, most webmail providers don't do that, because the expectation with email has been that the other person doesn't see anything until you hit "Send."
Unless the other person has a time machine, or you're using some unusual email provider and the other person is using the same provider, they won't even know that you've started to type a reply, let alone the contents.
It would be more of real time chat and not email (electronic "mail"), isn't it? with a API based json calls one could easily accomplish that but there would be adverse security violation in having this feature.