Why am I told the temperature of the mixture when I don't use it in the calculation?

K _p is only constant at constant temperature. If the temperature of the equilibrium mixture changes then so does the value of K_p.

Would the value of K_p change if the total pressure was different?

No. K_p would be the same because it is a constant at constant temperature. The partial pressures would, however, be different.

Why do we use K_p and not K_c for this reaction?

We could use K_c but it's easier to measure the pressure of a gas rather than its concentration, so we use K_p instead of K_c.