Is the ratio of HP to weight constant for identical air frames of different sizes? (See Scenario)
In an aircraft, engine power is used to overcome drag. Drag is related to weight, but not in an 1:1 ratio. 
There are several types of drag: 


Parasite drag: related to the surface area of the aircraft, incl. all the aircraft components that don't produce lift.
Lift induced drag: this is related to lift, which is related to weight.
Wave drag: we'll ignore this for low speeds.


(I'm simplifying things here) 

Total drag depends on airspeed. like this: 



When you scale an aircraft, things get more complicated. The weight scales with the volume of the aircraft. Your 1/6 scale aircraft won't weigh 1/6 of the original, but (1/6)^3.
Drag scales with surface area, so it'll be closer to (1/6)^2. I suspect lift scales with surface area as well. 

So as your aircraft gets smaller, its weight drops faster than its lift, so the wing loading gets lower. 

This also means the control surfaces become more effective, and the roll/pitch/yaw rates go up. 

This means your takeoff and landing speeds become lower. 

In practice, model aircraft fly at much lower speeds than their fullsize counterparts. So they're in a different part of the drag vs. speed graph shown above. This complicates the calculation of how much engine power you need for a model. 

If you made a video recording of a full-size aircraft and sped it up, you'd get a decent approximation of what the scaled aircraft would look like.