Been trying to figure that out myself. So far, been able to do this: prostitute yourself with the delivery service (the place you can sell your panties) 3 times, then you can do the brothel, do that 3 times, make sure you've had sex with Takua (or however you spell his name), find him and Aina at night near the church (took me ages to find this), this allows you to go out of the city at night, go to Koln mountains and a scene will happen, this is where I just got. May update this as this progresses.
EDIT: go to Koln mountains at night again, then next night, talk to Takuya, go to the remote house southwest of the hotsprings to have sex with Shouta and the others for the first time. I presume that this is the event the triggers you being able to summon them to have sex with, and that once you've done that (and possibly gone to the hot springs at night) you can finally do Duke's dammed banquet.
EDIT2: No, what you can actually do next is go into the castle at night, go to summoning alter place, and this triggers a scene with Abel and Takuya's group and Aina, then after that you can go do Duke's banquet. You can do the hotspring sex at night at this point, and possibly before you go to Duke's banquet, I don't know because I only tried afterwards. Oh, and at this point you can definitely start the scenes that involve summoning Takuya's friends. At this point, pretty much all sex scenes are unlocked essentially. The ones that aren't endings anyway. Speaking of endings, at this point, if you choose "we can't" in the ending scene with Takeshi, this results in Mai going back to the fantasy world, (mechanically much like what happens after you see an ending and it puts you back into the playable area), and when you try to leave the city, Camilla shows up and says that Takuya is waiting for you somewhere. He's in the Mysterious sewer, the spirits who prevented you from exploring there will be gone, and at the end you have a special boss fight, and once that's done, you get the sex toilet end.
Hope this improvised sex guide helps people out.