What do Murph and NASA do to save humanity after solving Brand's equation?

At the end of the movie Cooper arrives at "Cooper station", a giant habitable space station with a cylindrical interior which was near Saturn at the time--many such stations were built, and apparently when Murph solved the gravity problem (by finding a theory of quantum gravity with the help of the data Cooper sent her from the black hole, see my answer here from more on this), the surviving humans were launched into space aboard such stations, I don't think there had been any effort to settle humanity on a planet yet (see the answers to this question for speculations about why they hadn't yet tried to settle Edmund's world).

As for how solving gravity allowed them to achieve this, it was because the stations they had built on Earth were too massive to launch in the normal way, so their only hope of launching them and achieving escape velocity from Earth was to temporarily alter the gravitational constant in the space occupied by the Earth, so that gravity would become much weaker for a time and it would be easier to escape the Earth's pull. This is described in chapter 31 of The Science of Interstellar, which was written by Kip Thorne, the physicist who was the science consultant for the movie:

Early in Interstellar, when Cooper first visits the NASA facility, he is shown a giant, cylindrical enclosure being constructed to carry thousands of humans into space and house them for many generations: a space colony. And he's told there are others being constructed elsewhere.

"How does it get off Earth?" Cooper asks the Professor. "Those first gravitational anomalies changed everything," the Professor replies. "Suddenly we knew that harnessing gravity was real. So I started working on the theory—and we started building this station."

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How did it get lifted into space? The key, of course, was the quantum data (in my scientist's interpretation, the quantum gravity laws) that TARS extracted from Gargantua's singularity (Chapters 26 and 28) and Cooper transmitted to Murph (Chapter 30).

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Murph must have figured out how to reduce Newton's gravitational constant G inside the Earth ... In my interpretation, with Newton's G reduced inside the Earth to, say, a thousandth of its normal value for, say, an hour, rocket engines could lift the enormous colonies into space.

As a byproduct, in my interpretation the Earth's core—no longer compressed by the enormous weight of the planet above—must have sprung outward, pushing the Earth's surface upward. Gigantic earthquakes and tsunamis must have followed, wreaking havoc on Earth as the colonies soared into space, a terrible price for the Earth to pay on top of its blight-driven catastrophe. When Newton's G was restored to its normal strength, the Earth must have shrunk back to its normal size, wreaking more earthquake and tsunami havoc.

But humanity was saved. And Cooper and ninety-four-year-old Murph were reunited. Then Cooper set out in search of Amelia Brand in the far reaches of the universe.

First of all, Brands equation was already solved. However that equation alone was not enough and Brand needed some additional data from a Black Hole's singularity for Plan-A (humans safely evacuating earth) to work.

During the last scenes, we come to know that human race survived and Cooper recovers on a Space station, orbiting Saturn. So this was not Edmund's planet. At this moment we also come to know that till that point Amelia was still alone on Edmund's planet.

It can be assumed that there were other similar space stations with humans living on them. I remember there was a dialogue during the last part of the movie where the doctor tells a recovering Cooper that her daughter is alive but not here right now and that she would reaching here within 2 weeks. So it is quite possible that Murphy was on a different space station.

To sum up, all we know till the last scene in the movie is that human race from earth survived and they were living on space stations. Also, a habitable planet (Edmund's planet) is found by Amelia but till that point she was alone on it. The very last scene shows that Cooper sets off to reunite with Amelia on Edmund's planet.

When Dr. Brand senior was first trying to convince Cooper to pilot the mission, he asks him what does he notice about the Launch chamber. Coop looks around and says "This entire facility is a centrifuge. Some kind of vehicle? A Space Station?" Then Brand explains that Gravity is the key to everything with a long winded expository speech. As he phrases it "How to get a viable amount of human life off the planet?". Of course, it also requires a viable amount of food production.

Once they cracked the formula to change gravity, the rest is simple. Build giant space stations that allow for both a viable number of humans (at least 5000 genetically diverse individuals of breeding capacity mentioned in plan b), and the soil and atmosphere needed to support the station. Just a matter of engineering.

The data helped Murph solve the equation that allowed them to launch the space ship that you see towards the end of the movie orbiting the worm hole. The space ship was the one being developed in the NASA facility when Cooper and the others leave earth.

I dont think the idea was to save all the humans on the plant but rather to save the human species. So no, the entire human population was not present on that ship. Maybe they made more ships later, who knows.

Yes, the plant that Brand lands on (Edmund's planet) is habitable. Towards the end of the movie she is shown to set up base there. That is where the ship is headed.

Murph's equation enables creation of wormholes, control of atmosphere around space objects (creation of mini planets with atmosphere) , creation of inverse gravity space station as seen at end of film, lifting of huge mass from the earth, i.e. cheap space travel without fuel.

The concept of finding quantum info inside a black hole without being having yet the technology to control earth's climate, is science nonsense, i figure that much is obvious.