Chemical kinetics of a reaction, rate constant, and activation energy From the Arrhenius equation in kinetics of reactions,

$$k = A\exp{\left(-\frac{E_\mathrm{a}}{RT}\right)}$$

Which tells us about the temperature dependence on rate constant of a reaction.

Activation energy is dependent on the temperature (as it is the energy difference between the average energy of the reactants and threshold energy, and since the average energy of reactants depends on the temperature, activation energy also should depend on temperature) Is my assumption that activation energy depends on temperature correct?

Moreover, is there any case where activation energy is zero or negative? (like in the case of spontaneous reaction or $E_\mathrm{a} = 0$) I feel that whenever activation energy is zero, all the molecules undergoing collisions should form products successfully, but this case won't occur.

But when we substitute $E_\mathrm{a} = 0$, in the Arrhenius equation, we get the result as $k = A$, i.e. rate constant will be equal to pre-exponential factor.
The rate constant and the activation energy tell us how fast the reaction proceeds (moles/sec), not whether it is spontaneous. The higher the temperature, the higher the rate constant and, for specified concentrations of reactants, the faster the reaction. The rate of a reaction always increases with increasing temperature (higher rate of collisions), so the activation energy is always positive.  As the temperature increases, the rate constant approaches the pre-exponential factor (E/T approaches zero). For some reactions, the activation energy is very low, so even at lower temperatures, the rate constant approaches the pre-exponential factor. The activation energy has a weak dependence on temperature, and, in practice, this temperature dependence is usually  neglected.