Without pictures it is hard to provide advice. I had two plain vanila pconcreate patios and I destryed them and poured a larger patio which I covered in rock and added a pergola. Here is a picture.

Structural support is important. Roof must be able to support snow load (if applicable in your area which means beam size determined by span and slope) or strong winds. All support beams to deck should transmit load directly over beams from deck to ground. House attachment does not carry load well unless attached to vertical posts or into house wall joist. Sample graphics suggest posts against house wall to carry roof --if so that is good with house attachment serving primarily to stiffen against sideways wind load. Since there are no diagonal trusses on post make sure there are cross pieces on bottom of roof section to limit flexing on side wind(twisting to collapse horizontal box)

If the doors are exposed to the elements, the most important issue regardless of whether you trim it out with wood, or stucco patch to match the existing, is that it is properly flashed/sealed behind it all before covering it up. I'll be installing a window in a stucco exterior wall in a few days and I asked how the homeowner wanted it finished, they chose to match up the stucco. A trick I've learned since matching the color coat is difficult at best on a project like this for many reasons, is to base coat the color coat,(typically white), and match the texture. Once that has cured/dried, take a piece of the original stucco color coat down and have some acrylic latex exterior paint matched up to the original stucco color and feather it in around the patched area. It's typically not a good idea to paint over traditional stucco concerning large areas, but using acrylic for small patched areas isn't going to hurt. Again the most important aspect of the project being flashing/sealing behind not interrupting the drainage plane

Thanks again for the valuable inputs. From the research so far, few things are clear: 1. Teardown will cost anywhere between $20K-$30K. 2. I could possibly remodel the house by adding another bedroom and a full bath. I am thinking that may be accomplished by using the screened-in patio space. (A1 area in the attached floorplan). 3. I do plan to stay put in this house for the forseeable future. My business and kids schools are close by (1 middle school and 1 elementary school). 4. Speaking to the village engineers, I know that I could do an expansion of the house by simply adding footers to the existing foundation. Sorry if these questions are slightly off topic and tangential: 1. Since I have to replace the roof anyway, is it possible to lift the ceiling/roof a bit? This will (I think/hope) will increase the amount of light entering the areas of the house, making it a bit more brighter. Currently, I have 8ft ceilings. 2. Currently the house has small windows and not big French doors. If I replace them with French doors or sliding doors, that should help with the daylight also? 3. How feasible is it to remove the pool and utilize some of that space for expanding the living space of the house? I understand this will adversely affect the resale value of the house. Attached is a floorplan.