Did I build this AC-coupled emitter follower correctly?
It's an amplifier circuit. That means it has an input and an output. 

Input is on the left and output is on the right. This is the conventional way to draw schematics, signal flow from left to right and voltage decreasing from top to bottom.

You appear to have drawn it correctly.



The first two parts of the question are looking at the DC operation points and since capacitors block DC it doesn't matter if they are floating or not. For the third part you may wish to test with a frequency source connected to the input.

You almost never tie an output to ground as it might overload the output transistor(s). Typically you would measure the output into a specific load resistor.