Understand the Acceleration Due to Gravity Formula and More!

A question regarding equations of motion
Suppose I drop a body from a height $h$. The initial velocity is zero. Now it reaches the ground and stops and finally becomes zero. So now u=0 and v=0. If I use the equation of motion $v=u+at$ to find value of acceleration and input u=0 and v=0 I get a=0 even though it is clearly wrong and there was acceleration due to gravity. So what is going on here? Where is my understanding going wrong?

Free falling of object with no air resistance
Why does an object with smaller mass hits the ground at same time compared to object with greater mass? I understand the acceleration due to gravity of earth will be same but won't the object with greater mass will fall faster?

Acceleration of Rain drop
Under the heading 'Points to remember',my textbook states:
It is possible that an object can be increasing in speed when its acceleration is decreasing as in case of a raindrop.
I could not get why acceleration of the raindrop is decreasing. Should not it be constant as acceleration due to gravity is constant? Or there is some misprinting in the book?`Please help.

Acceleration of the Moon due to the Earth's gravitational pull
If the Moon has an acceleration of 0.0027 m/s^2 with respect to the earth when in orbit, and it is stopped for an instant and then released, will the acceleration towards the earth now be different? If yes, will its acceleration towards the earth be constant or variable over time?
(I'm talking purely about Newtonian gravity and pre-relativistic physics, and ignoring the gravitational effects due to other heavenly bodies.)

Vertical motion of an object
We define acceleration as $-<!--\; -\; -->9.8\frac{m}{s^2}$ for an object that is thrown upwards due to gravity. My question is when the object reaches maximum height at the point where it is about to turn around downwards, will the magnitude of the acceleration still be $9.8\frac{m}{s^2}$ or will it be greater than this as to help it turn around?

Question about acceleration equations that can be used for a single object
Can the centripetal acceleration equation $a=\frac{v^2}{r}$ and the acceleration due to gravity equation $a=\frac{Gm}{r^2}$ be used interchangeably to find the acceleration of a single object?

Acceleration in the rate of expanansion of the universe due to weakening gravity?
Could the acceleration in the rate of expansion of the universe be due to the weakening of gravitational forces, as the distance between objects continues to increase?

Normal reaction - force without acceleration
When a body lies on the surface of the Earth it is under the influence of gravity. The force on the body due to gravity causes it to exert a force on the ground and the normal reaction acts in the opposite direction causing the resultant force on the body to be zero.
However, how can the body exert a force on the ground when it does not have any acceleration? Since force equals mass times acceleration how does a body without acceleration experience a force?

Motion of a pendulum
The equations of motions for a simple pendulum is given by
$$\stackrel{¨<!--\; ¨\; -->}{\mathrm{\theta <!--\; \theta \; -->}}=-<!--\; -\; -->\frac{g}{\mathrm{\ell <!--\; \ell \; -->}}\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->}),$$
where $g$ is acceleration due to gravity and $\mathrm{\ell <!--\; \ell \; -->}$ is the length of the pendulum's string. Notice that the differential equation is of second order, does this mean that if I solve this equation numerically, the numbers that I get refers to the change in the velocity of the pendulum?

Design Lab for students
As a part time instructor, I was given a design lab that I would create and implement for students. For acceleration due to gravity that is not replicating overused labs related to measuring period vs length, angle or mass, please advise.

Why does it hurt more to fall from 10 stories, than 1?
$F=ma$. Falling from any distance, mass stays the same, and acceleration due to gravity stays the same. So, why does it hurt more, the longer you fall?

Does $v^2=v_0^2-<!--\; -\; -->2gh$ work if the positive axis is up and the initial velocity is down?
In the situation where the positive axis is up, the acceleration due to gravity is $g$, and the velocity is represented by the equation $v^2=v_0^2-<!--\; -\; -->2gh$. This works great if the initial velocity is upwards. But say, what if the initial velocity is downwards?. While we all know intuitively that the acceleration due to gravity would increase the speed (and thus $v^2$ would increase), this equation says that the speed (and thus $v^2$) would decrease. How is this justified?

Determine the impulse required for a projectile to reach to specific height with a variable initial velocity
Assume we have a projectile which we want to shoot straight up into the air, such that we are only working with the y-component, what formula can be used to determine the impulse required for this projectile to reach a certain height?
I was able to derive the following formula assuming no initial velocity:
$$F\mathrm{\Delta <!--\; \Delta \; -->}t=m\sqrt{2gh}$$
with $F$ being the force applied to the projectile, $\mathrm{\Delta <!--\; \Delta \; -->}t$ being amount of time the force is applied, $g$ being the acceleration due to gravity, $m$ being the mass of the projectile, and $h$ being the height that the projectile must reach.
Is there way to derive this formula to include an initial velocity? My attempts have all failed.

The pilot of an aircraft wishes to head directly east, but is faced with a wind speed of 40 miles per hour from the northwest. If the pilot maintains an airspeed of 250 milesper hour. What compass heading should be maintained? What is the actual speed of the aircraft?

a lamp hangs vertically from a cord in a descending elevator that decelerates at $1.8m/s^2$. (a) If the tension in the cord is 92 N, what is the lamp's mass> (b) What is the cord's tension when the elevator ascends with an upward acceleration of $1.8m/s^2$?

Non-Constant Acceleration due to Gravity
Recently, I had the first physics lab for my university physics course. This lab was fairly simple, as we were merely using a computer and a distance sensor to graph the position, velocity, and acceleration of a cart as it moved along a linear track.
One of the situations we captured data for involved starting the cart at the bottom of an inclined ramp and giving it a push upwards. As expected, it rolled up, came to a stop, and then came back down the track to its starting position. The position-vs-time graph was essentially parabolic, the velocity-vs-time graph was essentially linear, and the acceleration-vs-time graph was essentially linear. So far, so good.
At this point in the lab, the instructor pointed out that, if the data was examined closely, the acceleration of the cart was greater while the cart was traveling upwards than when the cart was traveling downwards (approximately $0.546\frac{m}{s^2}$ and $0.486\frac{m}{s^2}$, respectively), and asked us to determine why in our lab report.
Now, gravity was the only force acting upon the cart, and thus it's acceleration should be a constant, at least at the scale our experiment was conducted at, so these results are completely baffling my lab group. So far, we have proposed the following ideas, but none of them seem very plausible.
Doppler effect on the ultrasonic distance sensor
Friction
Air resistance
Human error
The first seems highly improbable, and the last three are more obfuscation and hand waving than actual theories.
Why does our experimental data show the acceleration due to gravity to change based on the direction the object is moving?

Meaning of acceleration due to gravity
I found article in which it is said that it would be incorrect to say that g is not acceleration due to gravity but local gravitational field as there is no acceleration on a block placed on a table. Please can you explain this as I am in a school and I have read only that g is acceleration due to gravity and textbooks say this too.