Convection is the movement of surrounding medium according to density by which transfer of heat takes place. Whereas in option A it tells about the transfer of heat due to molecules inside conductor, which is conduction.

Radiation is the fastest mode of transfer of heat, because radiation travels at the speed of light, which is very quick. The slowest mode of transfer of heat is conduction because it takes place from particle to particle.

The sun heats the earth through radiation. Since there is no medium (like the gas in our atmosphere) in space, radiation is the primary way that heat travels in space. When the heat reaches the earth it warms the molecules of the atmosphere, and they warm other molecules and so on.

Temperature of black body should be more than zero. Obvious answer. If the temperature is not greater than 0 then there will be no radiation.

The rate of loss of heat by a body is directly proportional to its excess temperature over that of the surroundings provided that this excess is small.

K = power × thickness (m) ÷ area × ∆T
K = (J s^-1) (m) ÷ (m²) (K)
K = J m^-1 s^-1 k^-1

The mechanism of transfer of heat between two adjacent parts of a body because of their temperature difference is known as conduction. This is the definition of conduction.

Newton’s law of cooling is to be used at temperatures around room temperature.

Newton's law of cooling states that the heat released by a body with respect to time (or) the rate of heat released is directly proportional to the difference between the body's temperature and the surrounding temperature. 
dH/dt = k(T – Ts) where t = surrounding's temperature and T = temperature of the body 
Consider two bodies A and B, of equal surface areas, such that A's temperature is more that B's temperature and the surrounding temperature is less than both A and B. Then according to Newton's law of cooling A loses more heat to the surroundings when compared B during the same time interval. So, A will cool faster than B.

Conductors have free electrons in them whereas insulators don’t have any. Therefore, conductors conduct heat and electricity better than insulators. Therefore the value of thermal coefficient is good in case of conductors and less in case in insulators.