If the moon was rapid enough would it be able to orbit the earth from a close distance? If the moon was close in orbit that it's surface was like 100 km away from the earth's surface. And it had a large enough angular velocity will it be able to hold orbit?
If this was possible, is something similar possible to exist in the universe (the very close orbit)?
If this was hypothetically in effect (ignoring the damage that would cause to life on earth), would a person on earth, or a person on the moon tend to levitate, or gain weight on the far sides?
 A: You should do some hard assumptions, as no atmosphere, but the real problem is the so called Roche Limit, that states that at this limit, the tidal forces , difference of the gravitational potential between the face facing the Earth and the one opposite, is so large that rips the body (Moon) apart.Wiki
With the Wikipedia formula:
$
d = 2.44 R_{Earth} \left(\frac{\rho_{Earth}}{\rho_{Moon}}\right)^{\frac{1}{3}} \sim 
2.44\cdot R_{Earth}km  \left(\frac{5.5}{3.3}\right)^{\frac{1}{3}} \sim 2.89R_{Earth}
$
That would be well beyond the Moon in your scenario.
But despite everything, the Moon and the Earth survives, and people also, then you should simply equate the gravitational pulls of each body.
$Earth_{Surface} = m_{individual}g = 70 \cdot 9.8m/s^2$.
$Moon_{100km above} = G\frac{M_{Moon}m_{individual}}{(d+R_{Moon})^2} \sim m_{individual}1.44 m/s^2$
where $m_{individual}$ is your mass. Earth pull would still be $6.8$ times stronger. You would not levitate, but you wouldn't need to follow a diet plan. 
A: Yes, it is possible for objects to orbit 100 km away from the Earth. After all, lots of man-made satellites are doing so. It's also possible for a moon to be much closer than it is now. However, you must realize that the radius of the Moon is 1,700 km so 100 km can surely not be the distance between the Moon's center and Earth's surface.
However, if the Moon were this close, it would have severe consequences. Yes, a person would feel lighter on the side closer to the Moon and heavier on the opposite side but despite the Moon's substantial size, it would still change the weight roughly by 1 percent in both directions (because the Moon is substantially lighter than the Earth).
This difference would primarily manifest itself as the tidal forces. They already exist but they're much weaker than in your hypothetical world.
In your hypothetical world, the Moon's center is about 8,000 km from the Earth's center – about 50 times closer than it is in the real world. Because the tidal forces scale like $1/r^3$, this amplification would translate to $50^3=125,000$ times stronger tidal forces. Instead of 1-meter tides, one would get 100-kilometer megatsunami on both sides of the Earth. Moreover, the tides wouldn't have a cycle of 12 hours. They would get repeated every 45 minutes or so.
Needless to say, 100-kilometer tides would splash everything on the surface of the globe. Lots of energy and angular momentum would be lost in this way, too. As a result, the Moon would soon either collapse to the Earth or flew away from the Earth.
