1. Yes, Cones 'N Scoops is probably an ice cream shop. (If it's not, the owner needs to rename it!) The name would be "Cones 'N Scoops."
2. Here, it sounds like it's probably a surfing-supply store. "Surf's up" is phrase meaning that conditions in the ocean are good for surfing. I have no idea if real-life surfers actually say "surf's up" at all, but it's a phrase that is commonly associated with surfing and surfers.
3. "Beach-pink" does indeed sound like it means pink like a beach, but I've never seen nor heard of any pink beaches. I searched for "beach pink" on Google Images, and all I saw were sunsets at the beach and one picture of pink flamingos on a beach. My guess is that the author thought it would be too confusing to say "beach-tan," since that sounds more like it's describing the tan that a person got at the beach. And "beach-beige" just sounds weird. So I guess the author went with "beach-pink."