As for your name and number showing up, to double check the obvious make sure the handset you tested on wasn't sync'd to Google allowing it to auto-populate the automated on-Device "Me" contact entry with your pic. If it was someone else's Google account synced phone, they have you in a Google Contacts entry w/ an email address, since that would pull in your pic and make it accessible to the dialer. That's their prerogative, not yours.
It was not my phone, it was a friends who had started from a fresh android 4.4 install. There was no google contact sync. Even if the contact sync was in place, it would be unlikely that the other party would have used my 5 year old YouTube avatar and populated it with both my email and phone number and full name. If my contact on his google side was populated from my gmail; as I have emailed the other party before, it shouldn't contain my phone number either - unless google matched it and shared.
Something went wrong. I think my account is an edge case and I was opted into a service (you are dickheads. Opt out is always evil.) and because I had not consumed enough other services my account was not given the option to opt out. Point remains, google matched my recovery number with my profile, which they said they wouldn't. I can't see any other explanation.
I'll repeat my alternate explanation more concretely: the name and the Youtube pic was auto-synced via @GMail, and your friend just manually added your phone number to it. It's now permanently in his Gmail account as part of Google Contacts, independent of anything done on Google's part.
Get your friend to delete his contact of you from the address book ("People") and see if it occurs again.
B) the option is NOT there if you do not have a google plus account. Do you need me to screenshot it? If you see the word "hangout" near the mobile number you have google plus account. I had one option, remove recovery number. There was no opt-out "help others find me" tick. Didn't exist. I have no "hangout" settings. Somehow my google account must be in some kind of limbo where I'm not actually using other google services.
Your only option was to purge the number from account recovery, which I have done. My name and "YouTube" picture still appear on android handsets without an explicit contact. It's probably cached like that iPhone to android iMessage case recently.