Something is going on at the very least where someone witnesses a feeling and they end up mimicking it unintentionally. Wearing the relevant expression itself can cause mild responses to fire off in relation to the intended feeling too, which is why some people advise others to 'smile more' over more than just appearances.
People see an emotion, and it ends up contagious through their openness to perceiving it. Even if not specifically 'mirror neurons' there is something culprit.
Fair enough, yes. From what I've read, mirror neurons seem implicated more evidentially in learning. "Monkey see, monkey do" stuff. Of course, the case can be made (and seems to be what is being made, obviously), that one develops (i.e. learns) the correlations of behavior and internal states (thoughts, emotions), which lend toward Theory of Mind, and down the line, empathy. Perhaps this is why...
Also, empathy and working with animals. Is it, you think, because animal emotional expression is less conflicted-seeming, than when interacting with humans, which can layer obfuscation and complexity?
I think it has to do with Body Language. People who seem sensitive to others feelings appear to also be more reactive to physical tells.
Mammals have a lot in common for it, it somewhat crosstrains without being as noisy and complicated as a person. It also likely plays into any mothering tendencies they have have within them.
...this holds relevancy to empathy, due to the correlative nature employed to develop it. This strengthens that spectrum of empathy, along the valence of this commonality you refer to. Despite animals having different lives -- by virtue of the fact of many things -- we find our commonalities outside the buzz of these preoccupations and the empathies for it, perhaps there is that solace to be found. It would also probably explain some of the reason humans like to anthropomorphize creatures and even objects, so as to understand and find commonality and belonging.
But to the point of the original post: does it warrant a separate distinction of "empath" or is it just human nature? In some respects, it seems to have the character of fish not realizing the water around them that they breathe and live in, or air-breathers the air they "swim in", or otherwise the invisibility of the mediums in which a creature lives. We don't go around calling people "air-breathers" or fish "water-breathers", except when we're talking about these characteristics in distinction to others (or their outliers), while most times how they are named or classified already imply that as "givens" to which they are involved.
Where do you place, if any, distinction between Affective and Cognitive Empathy?
Isn't Cog Empathy basically just sympathy?
A cursory search only brought up this, most relevantly. I would tend to agree with their equivalence, from how I understand them, so far.
Thrall to the Wire of Self-Excited Circuit.