Convolution of Two Impulse Signals
Your answer is right, assuming you posted the question right. 

But you better use the standard notation as Dilip Sarwate already indicated; $u[n]$ is the unit-step and $\delta[n]$ is the unit impulse. Then

$$ 0.5^n \delta[n-2] \star \delta[n] = 0.5^2 \delta[n-2] = \begin{cases} { 0.25 ~~~, ~~~n= 2 \\ 0.00 ~~~,~~~n \neq 2 } \end{cases}
$$

you can get the answer for the second case, exactly in the same way.