Here is the facebook link to the hardscaping company's photos

Here is a link that might be useful: Classic_Hardscaping

There is a flickr.com set of (too many) photos of the back yard and the link below. Tomorrow I'll edit to get just the pertinent pictures posted.

Here is a link that might be useful: flickr set

In order to tell if the dining patio is large enough for the d. set, you would need to show plan scale and compare with d. set measurements.

Having a pergola would depend on your level of interest compared to cost. IMO, the oval/round patio shape does not scream for a rectangular/square (as most are) pergola. Other shade and screening might be better/less expensive.

"...I'm thinking the far side, between the canopy of a crape myrtle and a neighbors large pine." From this description, I don't know where this place is. (The smaller notes on the plan are not readable; the plant material is unknown.)

if you surround flagstone with pebbles, I think you will lose the nice effect. I would explore if there is a groundcover that would work with the low light conditions.

general notes about plan ...

Excellent advice and thank you Yardvaark.

I hadn't thought about the pergola shape. Good advice! I suppose we can use the umbrella that can accompany the table.

The general notes I agree with completely. I was worried about that looping pinch. Consider that advice taken. I also agreed with the awkward space for planting, and the flow comment.

I don't have a scale plan yet. But I do have the yard dimensions from our plat, where I can see that the total North-South space available is around 30 feet from the near south side of the patio (by your green p) to the far North fence (where your yellow much word is) . I do have the table dimensions:
Table dimensions: 86" L x 42" D x 29" H
Chair dimensions: 25.5" L x 22" D x 33" H
The table image is uploaded.
I'll google room for table dimensions and see what comes up as a general rule of thumb.

I'm also wondering where to put the grill. The kitchen is in the middle rear of the house.

The plat is linked to below

Here is a link that might be useful: Plat

"I'm also wondering where to put the grill. The kitchen is in the middle rear of the house." Since grills are generally most conveniently located somewhere between the door to the kitchen access and the outdoor dining area, it's likely that the grill would go in the vicinity of where I show the chartreuse-colored double-headed arrow, where there is some blank wall space and depending on what the window situation is. This is a good reason why to especially have no pinching in that area.