If a star is twice as far away, is it true that it will appear to us 4 times dimmer? If this is true, can we assume that if we wish to view an object twice as far away while maintaining the same clarity, then the primary mirror of a telescope would have to be 4 times larger in diameter?
To your first question, yes. Ignoring any extinction, beaming effects, etc, the intensity of a light source that is radiating isotropically (equally in every direction) falls off as a function of the square of the distance between the source and the detector. With the telescope question, you are missing a couple of things here. One is that light-gathering ability (assuming un-obscured, un-attenuated light paths) increases as a function of the square of the diameter of the primary. Ignore pi (essential for computing area) for now, and ignore central obstruction etc and look at the relation of a 10" mirror (10x10=100) and a 12" mirror (12x12=144). You're getting a pretty substantial increase in light-gathering ability for a modest increase in diameter. You're also gaining resolving power with that increase, though that does not follow the "squares law".
Just to clarify, you need a telescope twice the diameter. Twice as far away is 1/4 thr brightness so you need 4x the area, but this is only twice the diameter.