Babychinos don't have coffee in them. It's throthy milk.
That's 2-1. Whoever told you was wrong I'm afraid.
My wife just texted me to say I'm right too. This doesn't happen often, so that's 3-1.
Cafes let dogs in at their own discretion, there is no law saying that you have to let dogs into cafes, pubs or anywhere unless they are blind dogs. Refusing a guide dog is illegal.