Yes, you can remove the edge pavers and add on with a different material.

You need to redirect the runoff with a drain system... I see you already have a retaining wall, you need to see where the water originates.

Have you spoken with a landscape architect?

If you are going to the effort, I would add more than just 2 feet.

Have you considered extending your roof to a pergola Or even replacing it all with a deck at floor level of the house, to eliminate the two steps down into the patio?

Thanks! We will be working with a landscape architect. Actually, this question came up because we are considering adding a small pergola with a slat wall to the right of the house (not really pictured) for privacy and maybe a tv.

The first search should be to contact local paver suppiers to see if you can match the existing material. For the area collecting water, pull those up and adjust the base to control drainage using a level or laser level based on a drainage plan.

The water is pooling because the yard slopes toward the house, and the patio is lower than the grass. Regrade to lower the grass around the perimeter of the patio, and grade a swale to extend around the side of garage. The slight dip towards the garage is not good either.

I don't see what another 2' around the perimeter, in the exact same configuration, provides. But you'd have to decide if you add soldier courses beyond existing, remove existing soldier course, remove cut pavers and fill out matching pattern all the way to a single soldier course edge again, etc.

I suggest redoing the entire thing.

1) You will have a difficult time matching the pavers. There are so many different brands and they will all fade/age with time exposed to the elements. If you add more, even with the exact same pavers, it will be noticeable. The "after thought" you are concerned about.

2) Drainage is an issue and that also suggests a full re-do. You are unlikely to get satisfactory results from just lifting, doing a bit of grading and resetting the problem area.

3) Finally, a 2 foot extension is not going to add much value or usable space. Go as large as you think you might ever want.

Re-doing the patio will also allow you to properly incorporate the footings and foundations for the pergola or overhead structure your situation is only hinting at now.

Do it right. Get your landscaper involved from the beginning.

I think since you already have issue with drainage the right answer is a new patio the right size . I know sounds expensive but biting the bullet once is better than trying to fix an issue that will not get better . That will aslo maybe allow a nice gazebo over the whole space too

Properly constructed patios should have stone below the pavers to eliminate issues. When building ours many years ago I created a French Drainage system by trenching a couple feet deep the entire perimeter of the patio, then leading to the swale at the side. With about 6” of stone covered with landscape fabric before topsoil there was never a problem. You could do this with the expansion.

As a DIY, I would trench the entire perimeter of the pavers as a swale about 6 inches deep and 3 feet wide. Lay a flexible 4 inch french drainpipe (with holes) at the bottom. Fill it with river rock and inset with large flagstones to the level of the pavers as a solid base for furniture.

Make another swale at the lowest spot in the corner leading out into the lawn and down the grade in a dip that would be mowable. Somewhere for the water to go. Reseed or lay sod to replace the lawn.