Why impedance between I/O pin and ground drops when Arduino is not powered
The Atmega328p (which is used in the Un[ (and actually all other AVR microcontrollers) has clamping diodes in the input hardware to protect it from overvoltage (above Vcc). When not powered, Vcc is at the same level as ground, so any positive voltage on an input pin is above Vcc. The protection diodes get conductive to protect the input hardware. Note, that the diodes can be fried easily, if a too high current flows through them. You should prevent that.
How to prevent it? By not doing it. You should never provide a voltage on an input of an unpowered chip (like any chip). If you really need to unpower the Arduino, you can prevent the voltage from reaching the input pin with a MOSFET.