If you put a roof over this area it will shade the bay window and the patio door. The interior will be darker. However, a PERGOLA is not a roof, so won't protect against rain. Be mindful of how much space you have and how much space can be taken up by furniture before things get crowded. Chairs offer more design versatility than a sofa. HERE is a Houzz article on furniture in outdoor spaces. As to wanting an outdoor fireplace, I would set it away from the house, and HERE is a link to Inspectopedia's rules on height of chimneys in relation to the roof.

There is a PRODUCT that allows a patio roof to overhang the home's roof. This prevents rain from getting through and allows more headroom and slope for the roof. These roof risers must sit over the top plate of the exterior wall, though, which means you might not be able to cover the side of the patio nearest the bay window. HERE is a photo of a completed roof addition. Otherwise, consider a standard PAVILION with an awning between the door and the pavilion.

Also plant some trees that will give you shade when you need it. Shade from a tree is cooler than the shade from an awning, because the trees evaporate water. I love the trees on the west side of my house. You can plant them in addition to adding an overhead structure.

Some options

Some ideas here http://insidetherosariansgarden.com/dw/45677/718nxa-fascinating-to-pergola/45631/

Don't forget that the hot west sun will come sideways onto the patio. Design your pergola with a cover on the west side, and/or plant something hedge-like. It can be a fast-growing shrub that loses its leaves in winter, since you only want the shade in summer. For example, a couple of large panicle hydrangeas like Limelight.