Assume I connect online from Japan. I host a domain and the host is based in USA. I logon to the webmail of this domain. I send an email to somebody in UK. Will the person receiving that email in UK see that the email is sent from a Japan IP or US IP?
it depends on how the webmail script is programmed. for example, i am looking at an email received from a "contact us" form i set up. this is not essentially different from webmail functionality. other than the email headers i specifically added, the mail server added headers that identify the host and mail servers by domain name and the mail server by ip address. within the body of the email i have included the ip address of the "web user", so the information is certainly available to the webmail script in your case. you can easily test this by sending yourself a webmail and examining the body and headers.