First the caveat - this is all experience on a turbo Miata, so much heavier than a 7. That said, I have a fair bit of experience with suspension tuning and high performance tires at autocross and on track. From your description of your intended use, I think Boxologist is right, the Toyo R888, Nitto NT01, Yoko A048 would be on the money. Toyo has reintroduced the RA1, which many prefer over the R888. My current track tires are NT01, and I find they're relatively civilized, very grippy (a subjective order of magnitude better than any street tire) and fairly long wearing (as track tires go).
Among real street tires, the current favourites are the Hankook RS3, the Dunlop Direzza Z1 Star Spec (not the Z101), the Bridgestone RE11 and the Kumho XS. The Azenis 615 are no longer a front ranked tire and are probably too stiff for a 7. I'm just coming off of a set of Kumho XS and I was very impressed with them. Easy to approach the limits, forgiving, very grippy and don't overheat as easily as the Falkens.
Nice list of toys Jim. I used to have an XK150S OTS when I was young and foolish - beautiful thing.