They simply tried to alter these.
For example:
In DMC1 the things were like this.
Force Edge: A memento of Dante's father, Sparda.
That doesn't necessarily mean that it was Sparda that left the sword for Dante. And in the end of DMC3 he has it along with his DMC1 clothes. So, about this nothing changed. It was simply us the fans who thought that the sword was left by Sparda.
Vergil thingie: Trish mentioned to Dante that he is the man who lost a mother and a brother to evil 20 years ago. About Eva, nothing has changed throughout the games. But they mentioned a brother and so we all thought that from the event in which Eva died, Vergil was kidnapped and turned into Nelo Angelo.
But it wasn't necessarily the case. In DMC3 Manga which takes place 1 year before DMC3, Dante is still under the impression that Vergil was killed during the assault that took his mother's life as well. In the end he meets him for once. And after one year the events of DMC3 come where he meets him 3 times.
So it is not so wrong to say that he actually lost him along with Eva. Before that they were always together and from the dialogue after Nelo Angelo's final battle we can see that they were pretty alike and close to each other. After Eva's death Vergil disappeared and for Dante the "Vergil" he knew disappeared also. Remember how he couldn't believe or understand what would Vergil achieve with Sparda's power.
The name of the shop is really an issue. If the anime is canon I can only conclude that since Dante changed it into "Devils Never Cry" because of Trish, he also rechanged it to "Devil May Cry" after the two of them decided to work separately.
If the anime isn't canon we can still wait and see the name of Dante's shop in DMC4