Your "question" actually includes several subquestions. Let's explore these:
If Dr. Brand (Sr.) had solved the equations, why didn't he use them
I feel the movie was a little sloppy in their terminology. What Dr. Brand Sr. had done was develop the equations. But just as in other research for mathematics, science and engineering; simply knowing the equations can sometimes NOT be sufficient.
We have developed equations that completely describe some things that are unsolvable or solvable only in specific cases. The example that I always think of are the Navier-Stokes equations that completely describe all fluid flows under all conditions.
These are mostly unsolvable - even using computational methods. Advances are made when new exact or computational methods to solve the equations for the various fluid regimes are developed.
What information needed to be observed from the black hole?
When developing equations to describe a physical system, often constants crop up. Sometimes the method used to develop the equations does not include any method of determining what the values of these constants should be.
If they had observations of a physical system for which those equations were valid, then those observations should be sufficient for calculating the value of the constants in the equation.
Without knowing those constants, the equations may not have provided enough information to be useful in solving humanity's immediate needs (leaving Earth).
Why did humans want to solve the gravity equation
The basic idea was that The Tyranny of the Rocket Equation would prevent evacuating any meaningful number of humans. Probably not enough establish a colony and make the population genetically stable.
The only solution was to develop a new means of leaving Earth that didn't involve the rocket equation. What's implied is that if humanity solved the gravity equation, we might find a means of leaving Earth (some sort of anti-gravity drive or equivalent). The book apparently explicitly states that this is what happens (and in the process, it breaks up the Earth).
How do we know the equations were solved?
I'm actually wrapping up several of your questions in this single answer.
It is implied that Cooper Station (and the complex in the NORAD base and presumably many others), show that the equations were solved because
- They're not designed like other aerospace vehicles (they use big heavy
- Apparently many of them left the Earth (we know of at least 3 - the one
that scooped Cooper Sr., Cooper Station, and the NORAD facility). I
assume there were many more.
- None of them had any apparent provisions for conventional propulsion
- At that time, the Earth had no other mechanism for lofting such massive
- Those structures were encountered by Cooper in orbit around Saturn (the
propellant requirements to put a massive structure there would have been
- I believe the book explicitly states that this is true.
**NOTE: Humanity has had the technology (Nuclear Pulse Propulsion) to loft such massive structures since the 1960s but neither the book nor movie make any reference to it.
Why else would we want to solve these equations
In no particular order,
- General human curiosity.
- The ability to go in & out of a black hole.
- The ability to launch stuff more easily from a planet & more
efficient space drives
- Better understanding of our universe
The italics ones are highly speculative. There's every reason to believe that we can't go in & out of black holes even if we have solved these equations, but the movie showed people doing it so - maybe.