At a certain point between the earth and the moon, the gravitational attraction towards the earth is exactly balanced by the gravitational attraction towards the moon. If a spacecraft is at that certain point, what is its distance from the center of the earth?

I don't know that much about maths, astronomy or physics as you'll realise. Does the moon have an eliptical orbit around the Earth? If it does then the ships distance from the moon would vary throughout the month or year.
Otherwise, the ships distance from the centre of the Earth when experiencing equal pull from both bodies would be at the point on the imaginary line connecting the two bodies that divides that line into two portions, the ratio of which is equal to the ratio of the Earths mass to that of the moon.
So get your encyclopaedia out and work it out for yourself because I'm too tired! Tiredness makes me grumpy.
Am I right?

The ship would be orbiting the Earth at a speed some fraction (same ratio or does pi get involved? ) of the moon's speed.

There would be a constant silhouette of the spaceship against the face of the moon, possibly even blotting it out completely depending on it's size! But if it was that large then maybe it could noticeably change the relative motions of the Earth and moon, i.e. speed of orbit and distance from each other.

The ship would also orbit the Earth much slower than if not under the influence of the moons gravity. It would not have to be accelerating around in a circular orbit to maintain the critical velocity for an orbiting body.

It's as if there were a gravitational link between the three bodies making them behave as though they were physically attached by solid poles, except that at least one of the bodies is spinning on an axis.

Am I talking garbage?!

Fury could take his encyclopaedia and look up Newton's law of gravity to find out that the ratio of the distances is not equal to the ratio of the resp. masses, but to the square root of that ratio.
Earth's mass is 5.9736*10exp24 kg, the moon's mass is 7.349*10exp22 kg, their ratio is approx. 81, square root of 81 is 9.
So, when in balance between earth and the moon, the spaceship will be 9 times further away from the earth than from the moon - and not 81 times.
The moon's orbit around earth is elliptical - shortest distance is 363,300 km, longest is 405,500 km, avarage is 384,400 km.
So, avarage, when in balance, the spaceship would be at 345,960 km distance from earth.