Dai-Shocker is introduced as "the union of evil organizations from multiple worlds". It wasn't composed only by Showa villains/AU versions of them, there were also Heisei members, although they generally seemed to be lower ranked than the Showa monsters.
The creation of Dai-Shocker is never directly shown. We know that Tsukasa became their "Great Leader" at some ponit (although we don't know if it was since the begginning of the organization as "Dai-"Shocker, or if he joined them later on) and he was the one who allowed them to start travelling to parallel worlds, expanding by allying with local villains and conquering the worlds. The Diend and Decade Drivers were weapons made to kill the Riders in Dai-Shocker's initial world conquest campaign, but the Diend Driver was stolen by Daiki. During his travels to defeat the Riders, Tsukasa somehow lost his memory and driver (which is found damaged in the first episode of the show). Tsukikage says that it might have been due to the impact of crossing the worlds, but he's kind of suspicious considering how he was trying to kill Tsukasa...
In All Riders vs Dai-Shocker, Tsukasa recovers his memories and joins them agani, but midway through the movie, Tsukasa is kicked out by Tsukikage/AU Shadowmoon, who says he has no need for Tsukasa anymore and he'll become the Century King.
Regarding Narutaki, like mentioned above, he wasn't a Dai-Shocker member and was shown to hate the organization several times. In Movie Wars 2010, he dresses up like Colonel Zol and gathers some Dai-Shocker remnants to create Super Shocker, eventually attempting to use them to kill Decade. In the end, he was knocked down by an explosion created by Doras, returned to his normal form, blamed Decade for everything and just disappeared.
The Shocker in Let's Go Kamen Riders had no actual relation to Dai-Shocker. Some Dai-Shocker members did pop up as silent suits and Decade's 10 faced demon actually got dialogue, but he was basically used as a replacement for Amazon's demon, and was only treated as the Geldam representative in the movie itself.
Now, Dai-Shocker is returning in Super Hero Wars, but we don't really know much about it at this point. It's using the original Shocker's emblem, rather than Dai-Shocker's, Narutaki's actor is playing the new Doctor G, although we don't know if it's Narutaki yet again, and Tsukasa is somehow back as their "Great Leader", but we don't know how that happened, why the villains are back again or basically anything about it aside from it existing.
Narutaki was never a Dai-Shocker member, he hated the organization as much as he hated Decade. However, after the Rider War started, Narutaki's hatred for Decade grew to such length that he tracked down the remnants of Dai-Shocker and created "Super Dai-Shocker".
The name is just "Super Shocker", not "Dai-" there.
