I agree that he should be looking into the other world he got summoned to, but at the same time there may not be much information on it. We know the gates to the dungeons connect to that world and the monsters come from there, but there is no indication that anyone was able to go anywhere else in that world. The only reason that Takeru and Karen were in the other world in the Demon Lord's castle he was at before is because the Demon Lord summoned them in some capacity. We also don't really know how much time has passed in the other world by comparison to Takeru's original world.
It could be that in the world where he was summoned it has been a few months to a year since his death or it could be 30 years. My guess is that his death in the other world was the trigger for the dungeons to show up in his original world...possibly something to do with the new Demon Lord we met a number of chapters ago.
I really hope the new Demon Lord isn't Edelweiss because that would just seem too obvious. To be fair, Sakura being Edelweiss is also too obvious, but seems to be more fitting here. Maybe Edelweiss was the one who cast some sort of spell to send his soul back to his original world and she followed along, but she hasn't "woken" up yet like he did. I am guessing death is what causes the awakening, so current time Takeru committing suicide happened and he died which brought OG Takeru back to his body. The same will have to happen to Sakura where she will die and then Edelweiss's soul will attach to the body and then boom we have Edelweiss again. It WILL suck if that actually does have to happen because I kind of like Sakura as a characterso losing that personality would be just sad.
Edelweiss is probably Calmia (the Demon Lord's daughter). First, she looks like her (sans the horns), and Takeru almost mistakes her too when they enter the Castle. She even reveals that "Pandora" can turn humans into powerful demons, and it's easy to imagine that Edelweiss would be devastated after loosing Takeru, and taking it herself. If I had to guess, Edelweiss took "Pandora" in order to revive Takeru, only to found out that his soul got swapped.
If anything it fits too perfectly, so I guess it could also be a misdirection from the Author. Either way I think she is lying about being the Demon Lord's daughter. We know she
does lie, since she also lied about killing Edelweiss (well, hopefully), and she isn't very bothered by her "father's" death. She also appears out of nowhere in the flashback to lead the demons, so my money is on she is just assuming this identity to lead the demon army in order to revive Takeru.
Also, she just renovates the Castle and uses it as her throne room for no reason... I think she is her.