When I send someone an email on Gmail, can they see the name I've set for them? Anything from your personal experience would be useful in answering this question.
There is a feature in Gmail called 'Show original' which allows you to see an email message in its 'raw/text' format, including its headers, encoded attachments, markup, etc. You can view a message in this format by clicking on the ▼ in the upper right corner of the message window and selecting 'Show original' from the drop down menu.
When viewing an email in this fashion you can see the name of the person who the message is being sent to. One message I viewed displayed this line:
To: Bill Hxxxxl <firstname.lastname@example.org>
To verify that a change to the name of the contact would also change this line (and that Gmail didn't have some secret sauce which tied a recipient's 'real' name to their email address) I added a middle initial to this contact's name and sent him another message. The To: line changed accordingly.
To: Bill J Hxxxxl <email@example.com>
So the name you give your contact is sent to them along with the email message. Will they see it? That probably depends on their email client, but I suspect that in most cases the answer will be yes.