Whenever I see an animal, usually a cat or a dog, do something really
cute, I feel a dull ache in the center of my palms.  This happens
fairly consistently, and I know I'm not imagining it.  A friend of
mine said she gets very slight throbbing pain in the back of her jaw
when she sees cute animals.  Why and how
do these things happen?

Your malady reminded me of someone afflicted with a variant of
Stendhal syndrome.  See
http://www.fact-index.com/s/st/stendhal_syndrome.html.  I draw this
analog only because there does appear to be precedent for the
occurrence of a physical reaction to the presence of certain
emotionally charged visual stimulii--in your case, "cute" behavior.  I
am neither a physician nor a psychiatrist, but I am certain from my
reading that it is of psychosomatic origin, i.e., it is doubtful this
is indiative of an actual organic, physical deficit.

Thanks for your research, but I'm not really sure that's the answer. 
After all, the linked webpage says that sufferers of Stendhal Syndrome
experience "rapid heartbeat, dizziness, confusion and hallucinations
when the individual is exposed to an overdose of beautiful art,
paintings and artistic masterpieces."  I mean, all I feel is a light
pain when seeing a cute animal.  I guess I could have a minor case of
Stendhal, but I wonder if anyone else has a better idea.