In Watchmen, did Rorschach's ever changing mask react to his emotions, or did it morph on its own?
No, it did not react to his emotional state. While the technology might have existed, and Ozymandias might have been able to make it, Rorschach's mask was actually part of an accident from his career working with a dressmaker.
From the Watchman Wiki:
Rorschach's mask (which he refers to as his "face") consisted of a specialized fabric, one that was actually two layers of fabric with viscous black and white fluids trapped between them. The fluids remained in constant motion, being affected by heat and pressure, however the black and white colors never combined to form gray. The patterns formed by the fluids also maintained a constant symmetrical pattern down the midline of whatever shape the fabric had been formed into.
Kovacs first discovered the fabric when he worked for a dressmaker. The fabric was originally made into a woman's dress and the marketing of such a product would be linked to Dr. Manhattan. The product's first customer was a woman named Kitty Genovese, who was unhappy with the dress, calling it "ugly" and refused to buy it. Kovacs thought differently however, and took possession of the fabric for himself, saying it was "very beautiful." At home he used heated instruments to shape the fabric into a full head covering, a face he "could look at in the mirror."
Curiously enough, there is thermochromatic paint that can change its appearance due to exposure to heat (from the breath of the wearer) but it does not yet mirror itself yet.
It's really left unanswered within the comic. It's most likely random, but as you can see by two of the answers provided, people have found patterns in it, just like a real Rorschach test. So it's symbolic of the idea that random patterns may provided consistent seemingly non-random reactions from people.