"This individual is clearly much younger than me and appears to be 17-18.".......Do you know the age of this girl/young woman? If not, you are assuming her age as some people look younger than their actual age. "Appears to be" sounds like an assumption.
"At the time I do not want to speak to him and would prefer to have any sort of relationship via email until I can forget the photograph of the girl on his lap.".........That sound too passive aggressive to me and isn't any way to approach this. You should actually talk with him on the phone or better yet SKYPE him if you can.
"I am aware my dad is entitled to do whatever he wants in his life however his lack of judgment to let me see something like that is giving me massive anxiety."........Althought it is concerning about your father and young women the decision what to be "anxious" over is in your control. FB.......yeah, it is an absolute double-edged sword. It's unfortunate he is posting his business on a social website for ALL to see, i.e. you and your friends.
Lastly, get all the facts about this before making assumptions and jumping to conclusions. You could be wrong. If you are correct, then simply voice to him how this is making you feel. In regards to your friends..... just let them know you aren't your father's keeper and you aren't responsible for what he does. Point out. I can imagine that this is embarrassing, but you don't owe them any long drawn out explaination about what he is doing, etc.
Hi there. Well, I feel your pain. LOL My dad married a woman two years older than me and three years younger than my sister after dating her for two months. Gotta love that. The new wife and I were in college at the same time! Yuck. I was a little embarrassed when they'd show up at my college for events (like a football game or something) to be honest. But . . . it is what it is. My friends and I just shrugged our shoulders.
Not much you can do about it. I'd stay off your dad's facebook page and if your friends say something to you --- reply "who knows. I don't get involved." and then DON'T.
By the way, I agree that it is hard to know the age of someone from a picture. And braces these days happen at any age. I'm in my 40's and can you believe my dentist asked me about braces a couple of years ago?. Now, when I was a young teen, I WANTED braces but now . . . not so much. Anyway, some do get braces after the tradition teenage years.
Anyway, we can't be responsible for our parents choices and just try to ignore the unpleasant stuff. good luck