Also, is it OK for this pergola or pavilion to be floating in the lawn (separate from the pavement under the main house)? Or should the house pavement get connected to the concrete under this new structure? Eg via some step stones or a concrete bridge. I’ve attached a photo of a stair step in my front yard as inspiration.

Oh also, we are thinking the size will be 20 feet wide and 15 ft deep. The back will be up against the property fence. The other sides will look out over the lawn, facing the house (to the left) and the playground (to the right). This picture shows you the lawn and playground it would look out onto.

Here are some basic ideas:

Waht are your plans for that area you seem to have a lot of entertaining space already that IMO could use some sun protection I need a bit more info What Doug shows to me is reminiscent of a folly in the old days the purpose needs to work with the design.

Doug, wow that is amazing to see a visual of your ideas! I love the surrounding trees in the mulch. I like your gazebo, especially the colors you suggested here, but I think I prefer a long rectangular shape so I could fit a main seating area (with furniture to fit 6-10 people), and also a table area with 2 long benches for sitting. I really like the placement of your gazebo, but am thinking of bringing it in closer to the house (to the middle of that border) to reduce the time to access it from the main house.

I'm sure you know that a pergola typically has beams for the roof which leaves the area below open to rain and snow and there will be less shade on hot days, while the pavilion has a solid roof cover. My choice would be dependent on the climate, whether there are other outdoor covered areas to use during inclement weather, and how it will be furnished (damp cushions are no fun).

If you have plenty of other covered areas, sometimes it's nice to have a bit more open sky.

Yes, you can float it and use hardscape paths to connect it to the existing hardscape.

JT7 that is great advice. I don’t have other covered areas in the yard. I live in Northern California in fairly warm/dry climate. No snow, and light rain. This yard gets a ton of sun since it’s south facing.
If I did a pergola, I would get a shade layer for the top, and potentially (not sure if this is possible) a solid layer that can open and close when it rains.

Could a wood or steel pergola look good, or would it clash with the house?

Here is a better picture that looks at the space straight on (marked in pink). This is roughly 20 feet wide and 8-10 feet tall.

I am going to make the grass area bigger by removing the rose bed that runs along the back (and put the pergola / pavilion in its place).

I am also removing the opposing rock wall (marked in neon green).