So how did it turn out? Fine. I made it with my mezzaluna, and not a food processor. I minced the basil, added the cashews, then the fresh garlic, then the freshly-grated parmigiano-reggiano. I added extra-virgin olive oil as I went, tasting for texture and liquidity. Be careful if the cashews are already salted to take that into account. Let me just say that making pesto with a mezzaluna is good (grainy), but labor-intensive.
After I made it, I remembered that the ready-made brand I usually bought in a small jar when I was living in Italy (Tigullio brand) did indeed use cashews (anacardi). And that was also good, and extremely labor-unintensive.
You should not cook or heat your pesto. What Italians will often do is to add some acqua di cottura (the boiling water from the pasta) to dilute the pesto. They will also add a noce di burro (about a teaspoon of butter) to each plate of pasta and pesto to enhance the taste and make it less tangled.