I'm putting an addition on the back of my house and want to remove a 13' wide section of the existing exterior wall on the first floor. The house is a 1918 cottage-style with sloping ceilings on the second floor. The wall I'm looking to remove is load-bearing. The current ceiling/2nd floor joists span 15' and are perpendicular to this wall. What depth of beam do I need and can I tie in the existing joists with hangers?

It is, I believe, unethical and almost impossible to design a beam without seeing the actual conditions at the site. Descriptions can be, and often are, lacking vital information needed to properly evaluate a structure and do the calculations required to design load-bearing members such as the beam you have asked about.

For example, there would be second-floor loads carried by this beam, but what about roof loads? There is no mention of the roof.

There is only one right way...have a professional (a good architect or structural engineer) LOOK at the actual conditions, take the required measurements, and design the beam properly. Not a place to be looking for short cuts...