Temperature and heat are not the same thing. Temperature is the measure of average kinetic energy of a given system while heat is its total energy (which is made up of more than its kinetic energy).
Take boiling water for example. As it reaches 100 degrees Celcius, any heat energy absorbed by water molecules contributes to a change in its potential energy (specifically, the breaking of any intermolecular bonds and whatnot). That is why there is no change in temperature during a phase change as the input energy does not contribute to its kinetic energy. When ice melts into water or when water evaporates, you will not see a change in temperature until the phase change is completed.
So in a quick summary, an input of heat does not necessarily correspond to an increase in temperature.
Edit: Didn't see your other questions.
B) This is where the concept of specific heat capacity comes in. Every substance requires a different amount of heat energy to increase its temperature for a single unit per unit mass. More heat energy is required to increase the temperature of a substance with a high value (such as water) compared to a substance with a lower one. You should be familiar with the equation:
C) A temperature is a measure of average kinetic energy. So if two substances are at the same temperature, then their average kinetic energy must be the same.