Can the recipient of my emails know my IP address?

Send yourself an email (at a non-gmail email address) via gmail and read the "full header" or "message source", especially "X-Originating-IP".

Compare this with your own external IP address (google [my ip is]).

The header of an email which I sent from my own gmail address to my own gmail address ("Show Original") did not contain an originating IP address, or any IP address except a reserved address in the 10.x.x.x range (in other words no IP address that would be identifiable as mine).

Then, I sent an email from my gmail address to myself at another email provider. The header of this email had an X-Originating-IP address which resolves to Google, Inc. It contained several IP addresses, none of which were identifiable as mine.

In appropriate cases a subpoena, search warrant or other lawful process directed to Google, Inc. would, of course, unmask your IP address and reveal your general location (and more legal process would force your ISP to reveal the IP address subscriber's name and address).

Your IP address was recorded by Google when you sent the email, and by God knows what other spooks inhabit the interwebs between you and your ISP, and between your ISP and Google, but it was not sent to your mail recipient(s).

If you use Gmail in as webmail (by sending emails on gmail.com), your IP address and hostname will not be leaked.

But if you use Outlook or Thunderbird like mail client to send gmail by using SMTP, then your IP address and hostname will be travel with your email.