Thermal grease / adhesive: Higher Conductivity & Lower Impedance or vice-versa?
Higher thermal conductivity (lower thermal resistance/impedance) means less temperature drop, all other things being equal. That is what you want as it will minimize the heating of the semiconductor die by minimizing the temperature difference between the die and ambient. 

The purpose of thermal grease is to fill the small (air) gaps between surfaces that are not perfectly smooth. If the surfaces were ground smooth and flat to a mirror finish, thermal grease would be of limited value as it would space the surfaces apart. Very smooth metal blocks such as gauge blocks used by machinists and in metrology labs can be stuck together simply by twisting (wringing) them together, and the gap is negligible. 

I have doubts about materials that may contain solid particles such as toothpaste- even if the thermal conductivity seems acceptable, the minimum layer thickness may be too high. If they're water-based, the water will dry out leaving air gaps. Grease has high viscosity but can be squeezed quite thin, and a good grease will not dry out.