Are total number of products purchased always correlated w/ total sales ($)? Would it be possible for this relationship to be uncorrelated if a lot of customers buys a lot of inexpensive product that don't contribute to sales revenue and then some customers buy fewer but expensive products?
 A: Sure thing. The correlation can be positive, close to zero or even negative! Suppose a business sells just two products. Product $A$ which sells for $\$50$ and product $B$ which sells for $\$0.01$. Now we consider $100$ transactions, consisting of customers who purchase some combination of the two products. The first $50$ customers (blue dots) purchase mostly product A and the second $50$ customers (orange dots) purchase mostly product $B$. The correlation of this data is $-0.45$. This is a classic example of Simpsons paradox.

Moral of the story: Always plot your data, and remember that correlation doesn't always tell the whole story. 
A: There certainly are scenarios where there can be negative correlation. For instance, if there are two clusters of customers, one making a few large purchases and another making lots of small purchases that add up to less than those of the first group. Perhaps for some reason someone decides to form a conglomerate that owns nothing but a car dealership and a grocery store. Then this conglomerate will find that the people going to the car dealership will have low count and high sales, while those going to the grocery would have higher count and lower sales.
However, in that case correlation is a poor summary statistic, as when people think of "correlation", they generally think of two continuous variables that have some relationship that's consistent over some range. In the hypothetical situation I described above, on the other hand, the negative correlation is purely an artifact of the clustering that is the effect of a confounding binary variable (whether someone goes to the car dealership or the grocery). Within each cluster, the correlation is likely positive (this is an example of Simpson's Paradox). 
Thus, saying "Count and sales are negatively correlated" is misleading, as it implies that increasing count in general decreases sales, when in fact there is a particular jump to larger count that decreases sales (in the example I gave, going between cars and groceries), and that one effect is so large that it's overwhelming the trend that otherwise exists for them to be positively correlated.
We could imagine a case without distinct clusters that still has negative correlation. For instance, suppose the equation count = 1/(average sale price)$^2$ is a good fit for the relationship between how many items a customer buys, and the average sale price among items bought by that customer. Then total sales would follow sales = 1/count and thus be negatively correlated.
As for there to be no correlation, that is theoretically possible from a mathematical point of view (if there's some parameter that sometimes results in positive correlation and sometimes negative, then there's some crossover point where it's zero), but from a practical point of view it would a rather large coincidence for everything to balance out perfectly.
