Is it physically possible to laugh while being angry? I have heard that it is impossible to both laugh and be angry. I tried and wasn't able to do it, even with simulated laughter and anger.

Is it possible to laugh and be angry?
As with all questions about emotion, the answer depends on how you define "emotion." Long post ahead… :P

Anger as basic

On the one hand, we have basic emotion theories, most famously advanced by Ekman, Tomkins, and Friesen and rooted in the work of Darwin. Basically, these theories suggest that emotions correspond to circumscribed regions of the brain (e.g., the amygdala for fear) and respectively involve specific and consistent patterns of facial, behavioral, physiological, neural, and experiential responses.  

According to this view, anger is a basic, universal emotion that is constituted by a specific set of facial movements, with which laughter would be incompatible.  So this theory would not predict that you could laugh while angry.

Anger as constructed

The most recent theories of emotion are called "constructionist"--although their ideas date back to William James (late 19th century).  Constructionist theories have been most famously advanced by Lisa Feldman Barrett and James Russell.  This view deeply challenges the basic-theory assumptions that emotions are discrete, universal, and modular.  

Instead, according to one constructionist account (Conceptual Act Theory), emotions emerge when we make meaning out of our situated affective experiences (to put it very simply).  In this sense, emotions are just concepts that can be applied to a heterogeneous population of experiences/contexts that share some statistical regularity (e.g., instances of fear often share the presence of threat/danger).  

So, one instance of anger need not look like another (e.g., many different facial expressions, CNS/PNS patterns, etc.), but in general I'll know what you mean when you say you're angry. Not to mention, your concept of anger may even be different than mine.  This means that you might conceptualize some instances as "anger" that I would not. So you would experience anger in situations that I would not.  One such situation might be where you're laughing. If you thought laughter signaled anger, then you might infer from your laughter that you are angry. 

So this theory would predict that it is definitely possible to laugh when angry.  Although, the more precise prediction would be: it is possible to be angry when you laugh.

Conclusion

I realize these theories are slightly confusing/complex (especially the second one)! I've read at least 600 pages about constructionism, and I still have to remind myself what it says exactly (it gets much more abstract).

I'll be frank and say that I completely reject the "Anger as basic" account.  Basic emotion theories are hanging on by a thread at this point, in my opinion--there's a huge and still growing body of evidence against them.  So my inclination is with the latter prediction: you can certainly be angry when you laugh.