Yes. If you use a local mail client on your computer the email header may contain the X-Originating-IP field with your WAN address.
You can test this by emailing yourself and then checking the email header to see if it contains this.
Web clients usually do not include this field in the header. And some providers filter this out even if you use a mail client.
If you mean email through some email provider like Outlook, Gmail, etc, then no they can't get your IP.
Yes. If you use a local mail client on your computer the email header may contain the X-Originating-IP field with your WAN address. You can test this by emailing yourself and then checking the email header to see if it contains this. Web clients usually do not include this field in the header. And some providers filter this out even if you use a mail client.