a pergola or a solid roof? How big an area, to the end of the stone pavers?

What kind of fireplace? a fire pit? do you have a gas source?

What room in the house belongs to the tree windows? the door leads to what/where?

Is that the only access to the pool?

Add a wood pergola, though you'll have to skip the fire pit, safer in any case so close to your house.

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If your main goal is to get some shade, I would look at what you can do with Sun Sails.

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I would do a retractable awning off the higher roof so you don’t impact the windows. I think a pergola would new tricky to get right and would negatively impact window view from inside

First, if possible, move the dining space so it's outside the door for ease of serving. Then set the seating space where dining area was and add a cantilevered umbrella for the seating area. Seems this would give a better view of the pool activity. If shade needed for dining area outside the door, look at having a metal pergola with light control at the top. For ease of maintenance, I'd avoid a wood pergola there.

The swap of the dining for the seating is a must if wanting to see the pool and I do not think a pergola is going to work there a all. A fireplace ???? Why? It looks like you have kids so IMO a must to move the seating to see the pool I love cantelivered umbrellas thay are great for allowing sun control or sails those are bit more complicated but work

Fireplace or fire pits are loved by kids. We used ours more five years ago when kids were younger than we do now!