Average length and entropy of a code from a probabilistic source
$\sum_{x\ne 000} p(x)$ is the probability of at least one 1 in the word. It's easy to break it according the number of ones ($\#1$) in $x$
$$=3 \Pr[\#1 =1] + 3\Pr[\#1=2] + 1\Pr[\#1=3]= 0.488$$

Which is indeed equal to $(1-(p_0)^3)$ as you wrote yourself.

The entropy is being computed similarly. Just plug in $\Pr(x)$ for any possible $x\in \{0,1\}^3$. BTW, your formula is written incorrectly (I assume it is a typo). Should be
$$ H(X) = -\sum_{a_i} p(a_i) \log_2(p(a_i)).$$