Because if the mass hits the ground or floor, it is no longer providing a source of energy for the discs rotation (no longer providing a torque to drive acceleration), and our expression is based on converting the GPE of the falling mass into the KE of the disc. 

We don't, if the mass hits the ground the disc still rotates and we can still time this so no problems at all here.

Only because, if the mass hits the ground it could damage something under it.

Because, if the mass hits the ground the bouncing will add extra energy to the disc

Because, if the mass hits the ground it will actually steal/remove energy from the disc.

actually we want the hole farther rather than closer so that the disc can accelerate to its ideal speed before we start timing and this will give us a truer period.

the farther away the hole starts the more work by friction in the axle matters

the farther away the hole is, the more delayed the start of the timer is and during this time the disc speeds up, where as our expression for the moment of inertia in terms of the time to rotate once assumes it starts from rest. 

the farther away the hole is, the more likely it becomes that the mass will hit the table as it descends.

no reason, regardless of where we start the gate will time one full rotation and that is what we seek.

It isn't, the direct flight would just need the potential energy of the separation between the Mun and the spacecraft,  it is just harder to time things for a direct flight that's why we don't use it.

Because, the transfer orbit makes use of existing potential energy of the separation between the Mun and the spacecraft and needs only to add a little more energy as a result to get to the Mun's orbit.

Because, the transfer orbit makes use of existing kinetic energy of the spacecraft and needs only to add a little more energy as a result to get to the Mun's orbit.

It isn't, the direct flight would just need the potential energy of the separation between the Mun and the spacecraft, but it is not possible to fire the engines in the direction required.

Burn away from Mun

Burn perpendicular to the plane of your orbit.

Burn retrograde

Burn prograde

Burn towards Mun

Burn directly towards the planet

Burn perpendicular to the plane of orbit

Burn retrograde

Burn prograde

Burn directly away from the planet