Does a Contacts Nickname transmit to anyone else; iMessage or email?

I had to laugh - but there's a serious question here, for sure.

Does a nickname in your Contacts show to anyone else?

Empirically, I would say no - neither a nickname you give yourself nor any other contact will be transmitted.
My partner & I have each other listed in Contacts with the nickname "ICE" [In Case of Emergency] yet it doesn't show up in any messages or emails we send to each other.
Similarly, my own nickname of Tetsujin is not seen by others in any communication I send.

I'm pretty sure you're safe asking Siri to call you Sugar Daddy… unless she does it in public, or if you ever intend to pass your vCard to someone, because it certainly will be in the exported card ;)