How did Murph know about Edmunds?

The only way Murph could know is when that information came from Brand herself. Therefore we must assume that the quantum data Cooper send to Earth/Murph was also able to fix the one-directional communication problem through the worm hole.

Well, you might also ask, how did Murph know Dr. Brand has reached the new planet at all. I guess Dr. Brand must have sent a message from the new planet, which informed others that she was there and she was alone. There might have been plans to reach out to her soon, only that Cooper could wait no longer and he immediately sets off for her.

I think the more straightforward answer is that:

1. Cooper had 2 weeks before he got to see Murph again, after being rescued and having recovered. In those 2 weeks he would have relayed their entire mission to NASA and told them exactly what had happened and how Amelia was heading to Edmunds' planet to execute Plan B.

2. When Cooper, Romilly and Amelia were voting on going to Mann's or Edmunds' planet, they knew that Edmunds' wasn't transmitting anymore, so they knew that there was a possibility that he might be dead or in hibernation.

3. When Murph is telling Cooper to go to Edmunds' planet for Amelia, she says, "Maybe right now she's setting in for a long nap.." - which implies that they don't have a word from Amelia Brand yet and the one directional communication might not have been fixed after all. So this is all conjecture.

I had to think about this too; your going to kick yourself because it is so glaringly obvious :) 91 years have passed since Cooper left. There was no time dilation on Edmund's planet. Edmund is dead if not by accident then by old age. She was not "informed" that Edmund died, it is just obvious because no one can live that long.

Of course she cannot be 100% sure about Edmonds's death, but you can easily assume he died since (if my memory does not fool me) he stopped sending any information (for a period of ~70 years, when Murph makes the statement), though his signal appeared to be still active

In the last shot when Amelie is seen in the new planet all alone, she is seen sitting besides a pile of stones with the 'dogtag/id' named EDMUNDS, pointing as a regard to fallen marine.

When Brand landed on Edmunds the time that had passed relative to Earth time was about 120 years - Brand shouted an estimate of this as they were being sling-shot around the black hole. So, Brand would have reached Edmunds and transmitted a signal. The spacecraft that Murph was now on could well have picked up that signal, which incidentally was around the time that Cooper and TARS were scooped up near Saturn.

The assumption is that the worm hole still existed by Saturn and that Cooper took the ranger to go through it and visit Brand again to re-assure her that everything succeeded. With enough fuel (this time round) he would not need to sling-shot Gargantua and simply arrive at Edmunds with no/little time loss. My guess is that the the spaceship that Murph was on is now going to head in that direction too in order to populate the new world OR now being advanced enough to make interstellar travel probably have lots of inhabited planets and Cooper went to pick her up from Edmunds.

When pilot Cooper wakes up, he will be debriefed, even though this isn't shown in the movie. Because of Murph's status, she will be given a short report of what happened, including the status of the other crew members. She knows that someone has to find Brand and all of the embryos, as there is a reasonable chance that they are alive. She knows that her Dad is the best person to do this as he is the only person alive who has been on the other side. She also knows that he won't leave her again unless she tells him to go and gives him some positive motivation.

In the beginning of the movie the NASA team informed that the hypersleep pod could help the pioneers survive for 10 years extendable to 20 years awaiting rescue. Dr Mann with superior skill extended that by using KIPPs power cell. Maybe Mann's planet was also quite close to Gargantua experiencing less time passing than Earth and even Edmunds' planet. Edmunds and all other pioneers though may have been presumed dead already after visiting Manns planet or even Millers due to the sheer amount of time passed. As of Dr Brand it can't be known that it is known with certainty from Our Solar System/Murph that Brand is dead or alive or how her mission is proceeding. On Cooper's journey back from the Tesseract though he mysteriously encounters Brand through Time - something that we could assume Cooper spoke about at Cooper Station - a strong message from the 5-dimensional beings to focus on Brand.

One Explanation Must be Murb using the quantum data and solve both ways communication. And send data how to to it to Brand . So, she can send info that she was the only one survived and on the planet alone waiting for recues .