On which factor does the thermal conductivity depend?

For cooking the food,which of the following type of utensil is most suitable?

$Assertion :$ $A$ brass tumbler feels much colder than a wooden tray on a chilly day.
$Reason :$ The thermal conductivity of brass is more than the thermal conductivity of wood.

In Searle's method for finding the thermal conductivity of metals,the temperature gradient along the bar:

$1.56 \times 10^5 \ J$ of heat is conducted through a $2 \ m^2$ wall of $12 \ cm$ thickness in one hour. The temperature difference between the two sides of the wall is $20^{\circ} C$. The thermal conductivity of the material of the wall is (in $W \ m^{-1} \ K^{-1}$):

$A$ metallic rod of cross-sectional area $9.0 \, cm^2$ and length $0.54 \, m$,with the surface insulated to prevent heat loss,has one end immersed in boiling water and the other in an ice-water mixture. The heat conducted through the rod melts the ice at the rate of $1 \, g$ for every $33 \, s$. The thermal conductivity of the rod is ....... $W m^{-1} K^{-1}$.