Can I still interpret a Q-Q plot that uses discrete/rounded data? I have a data set with only discrete/rounded values in it. As a result, when I produce a Q-Q plot a "stair-case" pattern appears. Can I still interpret this just like a normal Q-Q plot even though it is a lot harder to do so? Are there any limitations to interpreting a plot like this?
 A: As you say, a staircase pattern is an inevitable side-effect of discreteness, but that is the only obvious limitation. 
The rule for quantile-quantile plots otherwise remains that departures from sameness of distributions are shown by departures from equality of quantiles. 
Here are some dopey examples. I simulated some Poisson distributions. In practice, it is clearly more engaging to look at real data of interest, but I focus here on the graphical principles. First, I show two samples from the same parent, a Poisson with mean 3. A nuance in the graph is the use of open circles as a plotting symbol together with jittering of points (addition of random noise) to underline that multiple pairs of quantiles are being overplotted at several positions. The line of equality is shown as a diagonal, as is common on quantile-quantile plots. 

As a minor variation, here is a quantile-quantile plot for a sample from a Poisson of mean 3 and one of mean 4. The mismatch between distributions is evident. 

Such graphics is, or should be, easy in any well-developed statistical software. For those interested, here is the Stata code used to develop the examples above: 
clear 
set scheme s1color 
set seed 2803 
set obs 1000 
gen y3_1 = rpoisson(3)
label var y3_1 "Poisson mean 3, sample 1"
gen y3_2 = rpoisson(3)
label var y3_2 "Poisson mean 3, sample 2"
gen y4 = rpoisson(4) 
label var y3_2 "Poisson mean 3, sample 2"
qqplot y3*, jitter(2) ms(Oh) 
label var y4 "Poisson mean 4" 
qqplot y3_1 y4, jitter(2) ms(Oh) 

Quantile-quantile plots are also often better on transformed scales, but that is true of continuous (or not rounded) variables too. For counted variables that include zero, square roots are most common but cube roots can be useful. Otherwise logarithms remain the most useful transformation for positive discrete or rounded variables. 
Incidentally, quantile plots also work well for discrete and rounded data. (Quantile plots, for single distributions, can also be thought of as quantile-quantile plots with reference distribution a standard uniform: a reference line of equality is thus typically not helpful.) 
Here is a display for the auto data bundled with Stata: 

There is quite a range of variable types here: values reported as integers but all distinct, values that are measurements but in practice highly rounded, an ordered scale (1..5), a binary variable that is 0 or 1, etc. Naturally quantile plots take any numeric coding literally, but otherwise they are intelligible and even informative at showing variables known to be discrete and variables continuous in principle but quite rounded in practice. The particular quantile plots here are drawn to maximise their family resemblance to box plots, as cumulative probabilities of 0(.25)1 are labelled on the horizontal axis and corresponding values are labelled on the vertical axis. For more discussion, see Cox, N.J. 2012. 
Axis practice, or what goes where on a graph. Stata Journal 12: 549-561
 
