Im prepared to take a few stone throws and say fur characters are one that often lacks creativity, causing many of them to become Mary/Gary Sues. Mostly its the "I have ears/tail" way of thinking, which is neat as a physical appearance, as humanity is often obsessed with anthropomorphic images in its history. (even a werewolf is a mix between human and wolf, and would classify as a furry)
However, that being said that there is great ideas of furry chars, some of them hugely successful. A good example is the Guin Saga novels, in which one of the main characters head is that of a leopard due to some strange mask that is stuck to him. It causes him to revert to more feral instincts at times, making the character a interesting complexity between man and beast. I find this area to be the strength of "furry" chars.
Sterotypes exist in large droves, however, such as the "'Happy go Lucky" cat girl, the tricky fox person, ect. That is not to say that the sterotype cannot invoke very fun and interesting characters, but they are the norm. Most particularly, I love "Meracle" from Star Ocean 4 because of her cat-like features tend to make up a lot of her personality. A good example is that in the Star Ocean series, people ride bunnies to get around (unique?). She sees it and remembers it as being "Big... Fluffy... Delicous and round" while licking her lips and raising her arms up almost in a dramatic praying manner, or later when your supposed to capture one she's running after it with a fork, knife, and bib, which creates humor in the stereotype and makes it enjoyable.
Mostly I find that people close their minds to the idea of furry characters, which I find ironic that they love werewolves when some African cultures have "werecats", and both are in all technicality fur characters mixed with shape shifting, (or human characters mixed with shape shifting, depending on how you look at it) Furry characters can be well written, and should be judged as any other character is judged, but it is not always so.
Now as to Elves, I enjoy it when people do different things with our concepts of elves. In Final Fantasy 11, elves are huge, lanky looking beings that live behind stone walls. They took away the stereotype of elves in the woods+magic and made them a military kingdom with knighthoods and honor, which was a pleasant change from the usual. In a roleplay im working on, my version of elves have darkened skin, live in a volcanic wasteland and ride giant moths. Strange? Yes. Unique? Hopefully so :)
Random, I played a surgeon and I know what you mean. The role seems almost lacking because your interaction is almost limited to your duty. It becomes difficult to jump into the story, and stand out from the rest. Look at war as a whole, we often say "Support Our Troops" but do little to mention the skilled medical performers that help keep them alive, or the people who provide them with weapons, ect. I suppose its lack of glory is what makes it a more difficult role.
Kestrel, you raise a good point that animals don't think like people, but in a known universe they might :). Mostly its a entertainment medium I think. Much like how Indian tribes give spirit and human personification of animals according to their traits or appearance. "Wise as a owl" or "Clever as a fox" comes to mind.