I am trying to figure out whether an email header contains the IP address of the host who sent the email.
The header is supposed to have a number of received: fields, which list that one mail server has received this email from another.
What exactly does the first received: header contain? IP of the host, who sent email or IP of the email sender's mail server?
Short answer is it depends. The headers are added by each server that handles the message along the chain and shows its own hostname/address, and who it got the message from.
If a client simply uses something like POP3/IMAP in Outlook, and sends to their ISP via SMTP, the ISP's SMTP server will add a header containing their own host name, showing that it was received from the IP address of the user.
If they're using something like Gmail or Exchange, then chances are the message will effectively come directly from the Gmail or Exchange transport servers, so you won't see the actual sender's address in the headers.
