I have built many shed type roofs over the years. A strong outboard support beam is important when the roof is not supported by a stud wall on a continuous foundation. Long time ago I worked out a offset H beam ( offset to fit the slope of the roof) that supports the roof very well, with no sag or deformation. I have never had a problem with it, a little overbuilt but the only problem with overbuilt is taking it apart if you change your mind lol.
Here,s some pictures.
So on to the roof, try to get in a few more hrs. today. If I get some more work done tomorrow I'll post again tomorrow.