What's with the weird definition for the Calorie?
A number of the Chemistry texts I've used or examined for adoption have indeed confused heat and thermal energy. They sometime refer to a body possessing heat. But Physics books are pretty unaminous that heat is the transfer of thermal energy and not a state property of matter.

I've never seen, in texts I've used, heat restricted strictly to the interface between the participants in the exchange. It refers to the total amount of thermal energy transferred between two bodies. It can also be used to describe how much energy was lost by one of the bodies in the exchange and how much was gained by the other. One way to reconcile this with your interface statement is that once the surface of a body warms due to contact with a hotter body, the higher temperature at the surface relative to the rest of the body will continue to cause heat to flow within the body.

As to the choice of temperature, it shouldn't near zero since this is where a phase change occurs.

Edit. In response to your edit, #2. Search for a table of "specific heat of water". Notice the specific heat is not constant as temperature varies.
I'll hazard a guess on the 14.5-15.5 part. Think 15 degrees. But, since an interval is required, let's begin a half degree before and end an equal amount after. As to why 15 degrees, hmmm. One answer to a recent question on the Physics forum says this is roughly room temperature. Pretty cold for me but maybe true when and where this was defined. (Sorry for no link, I'm not that fascile on iPad.)