Can you have a z score that extends past -5 or +5? Do z-scores run from negative infinity to positive infinity? 
 A: Yes and yes. For example, if you transform a vector with a hundred 1s and a single 2 to z-scores, the 2 will become a z-score of 10. Replace the 2 with -1 and it'll get transformed to -10 instead.
A: By $z$-scores we mean subtracting the mean and dividing by standard deviation. So to obtain $z$-score of five or more you need $|x-\bar x|$ simply to be greater than five standard deviations...
Moreover, from Chebyshev's inequality we know that no matter what is the distribution of your data
$$ \Pr(|X-\mu|\geq k\sigma) \leq \frac{1}{k^2} $$
so probability of observing $x$'s distant by five standard deviations from the mean is at most $1/25$. Even if you were thinking of normal distribution and the "three sigma" rule, then $z$-scores of $-5$ or $5$ would appear in less than $1\%$ of cases but with non-zero probability.
But you ask also

Do z-scores run from negative infinity to positive infinity?

In theory if $x \in (-\infty, \infty)$, then obviously $z = \frac{x-\mu}{\sigma}$ can be infinite. However in real life you cannot observe infinite values, so $z$-scores cannot be infinite. They can lead to huge values, but not infinite.
