The first part is correct. Increasing temp increases both rates. You're also correct that the temperature does not affect the energy of activation, since the energy of activation is a fundamental property of the molecules and transition states involved. Increased temperature increases the energy of the molecules, which makes it easier for the molecules to get over the energy barrier. But the barrier is still just as high.
There is one part of that first paragraph that I have a problem with. You keep referring to "the rate constant K." But in a reversible chemical reaction, there are two rate constants. There is the rate for the forward direction, and the rate for the reverse direction. In the case of increasing temperature, both the forward and reverse rate constants increase. Perhaps you're referring to the equilibrium constant, which is the ratio of the forward and reverse rate constants. Equilibrium constants are temperature dependent, and the temperature dependence is given by the Van't Hoff equation (basically, if the change in enthalpy of the reaction is negative, then the equilibrium constant decreases with increasing temp, if it is positive, the equilibrium constant increases with increasing temp).
Your second paragraph is partially incorrect. Catalysts increase both the forward and the reverse rate constants (this is known as the principle of microscopic reversibility). This is how a catalyst increases the rate of a chemical reaction. However, catalysts do not affect the equilibrium constant.
Incidentally, if you're not sure if something affects the rate constant of a chemical reaction, but you know that the rate is affected, here's how you can tell if the rate constant is affected:
The rate of a chemical reaction is given as
Rate=k[A]^m*^n...
where A and B are reactants m and n are the respective orders of those reactants.
So, the rate only depends mathematically on two things, the rate constant and the concentration of reactants. Soooo, if you know the rate is affected, and you know the concentration of reactants is not affected, then the rate constant has to be affected.