Welcome to TLF, glad to have you here. :thumbup:
You should consider improving the "shady corner" by turning it into a flowerbed for shade-loving plants, or just a mulch bed for easier maintenance. You're going to have a hard time getting grass to grow in that area. I took the liberty to edit your picture with some visualization of what you could have. Depending on preference, I would extend the mulch bed around the back of the porch. Not sure how that works for your living space.
I would mulch around the shed. Too much shade.
Why did you put down a fungicide? You've got a good case of crabgrass and goose grass that I can see in that 4th photo. Honestly, the best thing you can do for your existing grass, if you choose not to renovate completely, is to remove the weed pressure, and feed the bermuda you have. It'll do the rest in short order.