I think it will look much more finished with the edging on both sides. There are some examples in the houzz hardscaping idea photos.

Is there a softer colour you can choose. The red is so bright it detracts from the landscaping and your home.

I don't think it would add much on one side only. And if you do it on both sides, it'll be a tripping hazard.

The sample photo looks more like it's being used to edge a flower bed, not to anchor the walk. If you do all your beds like that, sure, go for it. But just the one? Nah.

The photo you found is a good example of using an edging. Having one on the other side might trap water. Notice how the brick was laid in the opposite direction, which I think helps stabilize it. I think it looks good.

That red brick is beautiful. People are paying big money for charming ones like that these days.

It is never a good idea to have any plant growing up against a house. Pachysandra is so invasive....I agree....it could grow into the brick.

You also need some height there.

I would remove the pachysandra and add stone or mulch and some showy flowers. Your landscaping is a sea of green and you could use some contrast.