Note that

$$ \lfloor \frac{a \% p + b \% p}{p} \rfloor = \lfloor \frac{a+b}{p} \rfloor - \lfloor \frac{a}{p} \rfloor - \lfloor \frac{b}{p} \rfloor$$

so your expression can be written as $F_{p_k\#}(x+p_k) - F_{p_k\#}(x) - F_{p_k\#}(p_k)$, where

$$ F_n(x) := \sum_{i|n} \lfloor \frac{x}{i} \rfloor \mu(i).$$

Now, the expression $F_n$ can be simplified by elementary number theoretic computations. Indeed, observe that

$$ \lfloor \frac{x}{i} \rfloor = \sum_{m \leq x: i|m} 1$$

and thus after interchanging summations,

$$ F_n(x) = \sum_{m \leq x} \sum_{i|n, i|m} \mu(i).$$

But by Mobius inversion, $\sum_{i|n, i|m} \mu(i)$ equals 1 when n,m are coprime and 0 otherwise. Thus $F_n(x)$ is nothing more than the number of natural numbers $m$ less than $x$ that are coprime to $n$. In particular, this gives

$$ F_{p_k\#}(x+p_k) \geq F_{p_k\#}(x)$$

and

$$ F_{p_k\#}(p_k) = 1$$

which explains your numerically observed phenomenon.

The quantities here are somewhat reminiscent of the quantity $\pi(x+y)-\pi(x)-\pi(y)$ that occurs in the second Hardy-Littlewood conjecture (which, by the way, is widely believed to be false). Indeed, the quantity $F_{p_k\#}(x)$ (that is, the number of natural numbers up to x that have no prime factor less than or equal to $p_k$) is more commonly denoted $\pi(x,p_k)$ in the analytic number theory literature.

EDIT: I also encountered some related expressions and inequalities in my paper http://arxiv.org/abs/0908.4323 (and with more advanced versions of these inequalities also in this paper of Granville and Soundararajan: http://arxiv.org/abs/math/0508361 ).