I'm glad to hear the guy has agreed to simply replace or repair the one rotted joist you found and to lay the circa 5 square meters of the floorboards you've purchased. If the £800 he's quoted includes sanding and finishing the entire floor spanning the lean-to and living area then that price may be reasonable. If it doesn't, then he's ripping you off.
Sadly no, it's just the joist and floorboards being put down. I never had this issue where I used to live, now I'm back in London it's really difficult finding someone. One other came and wanted £300 but then I googled his name and discovered he had run off with peoples money.