From your description at least, I'd call that a housing. The shoulder would be part of the tenoned piece. A basic rule of thumb would be 0- 1/2" housings for beams that have little to load on them. And 1", maybe larger, for beams with floor loads, roof loads, etc.

A historic standard for the distance off the shoulder for pins is 1.5". That way a standard framing square can be used. If the joint is under tension, it may move away from the shoulder.

There really aren't any simple methods for the span on timbers. Partly because it depends on too many variables. There was a discussion on the TFG's Facebook page just over a month ago on this subject. There are a few online calculators. I wrote up an Excel worksheet.