Refining my thoughts and looking at your picture again, and seeing that you are not comfortable with modifying the existing house, I would suggest doing two openings where each window is. I'm guessing that there is a load bearing wall down the middle of the house with each window being a different room. Leave that wall between the rooms alone for now and just make the window openings bigger. Then you wont have to worry about any structural issues. I would still install a header over the openings, but that's super easy to do.
Remind me what you want to put in the lean to. If it's a tractor, how will you fit it under the lean to with such limited clearance? A lawn mower will fit under the lean to, and some other things, but with that roof being so low, you are very limited in what will fit under it. Figure the size of your lumber and measure down from the low side of what you want to go in there. If it's a tractor, measure down from the under side of the joists at least 6 feet away from the building, with 8 feet being more realistic so you can get around all four sides of the tractor.
Also do a drawing to scale on some graph paper that includes the thickness of the joists with the roof finished. If you are installing metal R panels, add an inch and a half for purlins if you space your joists every 4 feet and two inches for the metal. Hopefully you will still have room left for flashing. Or if you are going with shingles, add another inch at least for OSB and shingles on top of the joists, which now have to be a lot closer together to deal with how heavy shingles are and your span. Shingles are where the span table really matters.