for the orginal circuit -- you've saturated the device. heavily at that. as a result, the output will be close to the emitter voltage, which is close to the base voltage, which is close to the input.
there are a few different equations involved. For simplicity, I suggest AC coupling the input and output, as this simplifies the design. You don't have to worry about source impedance affecting the biasing, nor do you have to worry about biasing to get 0V on the output.
also, the output changes with no input connected because the biasing is affected by the source impedance. When a source is connected, the pin is a low impedance connection to an average of 0V. When disconnected, it is high impedance. This changes the bias voltage at the base, and thus the biasing of the output.