Would I be using the ty form if I didn't even know the person's name. Or is this talking to a child?
That really depends on the social situation. Normally, one uses the vy form for adult strangers, but in certain communities (certain sports and hobbies, perhaps religious communities,...) people call each other ty by default.
Why is this 'jak' and not 'co'?
I have jak= how in my notes
Try to remember the whole phrase. It is similiar to German "Wie heißt du?"
It is like "How are you called/How do they call you"
"What" also gets translated as "jaký" or "který" in Czech. Here, you could ask "Jaké je tvoje jméno?", but "Jak se jmenuješ?" is used more often when asking for someone's name.
Or literally "how do you call yourself?"
That is not really literal to me. I would accept that for Slovak "Ako sa voláš?" where volat is really to call (even by a telephone).
How would you normally answer that question? "Jmenuji se _" or "Jsem ____"
Either is fine, but "Jmenuji se " follows the question more directly.
Could you also answer "moje jméno je ___" ?
You could. But it is less natural as a reaction here.