Isolation
Do any of us truely live in isolation? World events, past and present, shape our lives. Pressures from society at large -- not to mention society at home -- also changes us. These are important points to keep in mind when writing, especially what would be considered "Slice-of-Life", or drama.
It can be hard to make your anthros -- heck, any character for that matter -- real. One of the big problems I see is that many characters seem to exist in isolation. I'll give you an example:
When I was younger (18 to be exact) I wrote a story I called "A Glass Half Full." I had written works of fiction prior to this. In fact, I believe my first 'anthro' work was written back in the third grade. However this was the first piece I considered good enough to keep. At that time my writing style was heavily influenced by online message board/chat room based RPG. I did manage not to write the work in that style, praise be.
In the years since, I have looked back at this and other works and said to my self, "What the hell..." Something was wrong with my stories (besides the fact that they weren't very well written.) Looking back, I now know what the problem was: they were written in isolation.
My furry characters were not explored culturaly, biologicaly, socialy, historicaly, or otherwise. They were people in fur coats. The characters were two-dimentional, to put it kindly. So I made revisions. Then I made revisions to my revisions. I have rewritten most of my older stories at least once. As it stands now, "A Glass Half Full" is in the process of being further fleshed out into what will probably be an online novel called "Cry, Wolf" (I'm up to about 60-70 pages now, and most of them worth reading. When they're all worth reading, I'll know it's ready for editing.)
You don't need to provide an entire universal context to your readers, but you do need to provide a world-view context for yourself, the author. This skill is one that will help you make your world, your characters, and your drama, more believable. Your short story deserves it. Your characters deserve it. Your readers will thank you.

Re: Isolation
You make some excellent points about what makes believable characters, but I would reserve the label "humans in fur coats" for a specific character design flaw, not lack of character design.
A serious flaw of many anthrofiction writers is to only name a character's species, and expect that is all the reader needs. It's a bit like writing a story about a superhero who does nothing super or heroic.
For example an author may write, "Simba, the young lion," but will do nothing to show us Simba is a lion. When the reader finishes such a story he has learned nothing about lions.
A good test for this flaw is to ask yourself, "How hard would it be to convert my anthro-characters to human?" If the answer is: "I need to change 'the young lion' to 'the boy' in three places and edit one sentence about fur color." Well then that is just pathetic. The story is not worthy of the label anthrofiction. If the answer is: "I need to rewrite the descriptions in the scenes where my anthro has point of view to replace all that stuff about how things smell with how they look; I need to rewrite the kissing scene; I need to edit the scenes about food because humans don't eat what my anthros eat; and I need to edit all those places where my anthros twitch their tails, flick their ears, lick their whiskers, and 'nose about' because if a human did all that it would be silly." Hey, that's pretty good anthrofiction!
But for the very best anthrofiction the answer will be something like: "Simba must be an African Lion. No other species would care about the things he cares about or act the way he does. Replace him with a human? If I did that the story would not make sense."
HOW TO FIX IT
I've managed to codify some of the things we authors can do to lift our anthros above the mundane.
Body language: Most mammals have more expressive ears than humans, and they have tails. Use them! This sort of thing is so simple there is no excuse for any author to not do this.
Diet: Not every animal eats exactly the same things that humans eat. Depending on the degree of anthropomorphizing their diet could remain similar to that of their wild cousins.
Behaviors: Natural animals have behaviors that are quite different from human behaviors. Partly this is due to having differently tuned "equipment" and differently tuned senses. E.g. bigger mouths and teeth, a sharper sense of smell and hearing, and a muzzle with thin lips may make kissing a different experience. Of course how different depends on the degree of anthropomorphizing and if your character has to fit into a foreign (human) society, how motivated they are to fit in.
Physiology: Besides having a different diet, a different reproductive cycle can radically alter how your characters interact with each other. Also consider the effect of having fur.
Senses: This is a biggie! Birds are visual. Fish operate on scent and vibrations, and some even see the world by detecting electric fields. Mammals, most mammals anyway, see the world through scent and sound. Most mammals are color blind--but then we are "nose blind" and can hardly imagine what it would be like to experience the rich world of scents. Our own Steve Abbott says that a human with total color blindness sees the world in sharp textures and can tell the difference between slight shade differences of red (for example).
A new technique for me is to try to write all my scenes, where an anthro is the point of view character, so everything is described in terms of the primary sense of the character's species rather than the dominant human sense of sight.
CONCLUSIONS
Writing fiction isn't about absolute accuracy, it's about creating an illusion. We don't truly need to know how an African lion (to use the example of Simba) thinks and feels, we need only create the illusion that the character is a lion and not a human.
This technique is a lot of work. You must know your species. The more you know the easier it is to write your characters and things like how their society might work just seem to suggest themselves. Which gets back to creating well-rounded believable characters--except in this case they are anthros.
Happy writing.
Scotty
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