I laid a similar walkway, except the edges are tumbled bluestone cobbles instead of brick. A handy or at least persistent DIYer could certainly do it.

Mine was dryset, because it lasts better in this snowy climate.

Cliff Notes version: You'll need to dig down about 7-8 inches, put in 4 inches of paver base/crushed stone/hardpack, tamp it really well, add a layer of landscape fabric, put in 1-2 inches of paver sand, tamp it really well again, then add you stones. Edging strips, either plastic or metal, keep the brick edging from heading for the grass. Put in your brick edging.

Now the really time- consuming part. Those irregular flags take a lot of careful planning and trial-and-error to fit together correctly. I estimate I averaged moving each stone 10 times before I got it right. You have to level each one individually, because some are thicker than others. This was also the most creative, enjoyable part of the job. But it does not go quickly ... at least not for this amateur. I filled in any large gaps with small stones from around the property, which gave it a more rustic look; you might not want this.

Then either rent a plate compactor or tamp the whole thing using a 2x4 under your tamper to make the stones level with each other.

Then sweep in sand. I used polymeric sand to keep weeds and ants out and to make it last longer.

Working with the irregular stones is nowhere near as speedy as working with uniform bricks or pavers, but the end results are worth it.

Exactly what chibimimi said with these additions: I use a worm drive saw with a diamond blade to trim the flagstones to fit precisely. It looks like the gaps are pointed with mortar after installation as well.

You mentioned leveling difficulties due to non-uniform flagstone thickness. I happenened upon an easy solution that I haven't heard of or found posted previously:

With the flagstones set on 2"-3" of decomposed granite base material, adjust the gaps to the width desired. Let the surface run wild for now. To level the stones, cover the entire project with decomposed granite and screed level. The peaks of the highest point of the stones should be level with the surface.

Next, dig down and pick up one stone, dump the base material into the depression, and replace the flagstone. Do this to the next stone as well, working just one stone at a time. You will have to repeat this process for every stone but it goes very fast because there is no trial and error, just pick up, dump and replace.

When you are done, the surface is level as the screed in the step above. It works because the base material that is sitting on top fo the stone is the exact volume of material that is required underneath it for the surface of the stone to sit at the exact height of all the other stones.