When I was doing my (Botany/Zoology) degree, I spent much of my time hanging around in our University museum helping the curator make relevant displays. The first thing about well-done taxidermy is that no guts are involved in the process. The deceased animal is carefully skinned and then the skin is treated to a) clean it thoroughly and b) keep it intact and unstretched ready to be 'stuffed'.
The 'stuffing' process doesn't mean the reconstituted skin is packed firmly with hobbyfill! The taxidermist has to have a profound knowledge of the animal's skeleton and musculature in order to get a good approximation of a framework on which the skin will be mounted. Sometimes, the skull will be left intact and its contents removed, other times it will be desiccated or freeze-dried. The main aim in the preserving process, whatever form it takes, is to remove decomposing bacteria so the skin will not rot over time. Also, it has to be insect-proofed so that Dermestid beetles (Museum Beetles) don't get in and chomp away at the connective tissue which holds the skin together. For larger animals, the framework might be made of wood or aluminium. For smaller ones, chicken wire or something similar works well.
Birds are extremely hard to 'stuff' because of their thin, supple and easily-torn skins. The feathers will fall entirely out of a preserved skin unless you take immediate steps to 'set' them. There's nothing very attractive about a bird skin with half its feathers gone!
While I was at Uni, our Zoo. department got a freeze-drying machine. I can't remember the principle behind it, but you would put your (dead) specimen into its stainless steel chamber and turn it on. After the desired cycle (a day for a reptile - large snake or lizard; a week for a larger, moister mammal - rabbit or cat), you could remove your specimen. The innards would be completely desiccated so they rattled around inside (could be easily removed through a slit in the abdomen or near the anus). Best of all, your specimen would be completely lifelike and in whatever pose you'd left it in.
This was how our Museum curator (lovely chap by the name of Glenn) came to fashion a rock band made out of pestilential Cane Toads. It was hilarious! Each toad had a black wig (made out of Glenn's donated ponytail) and wore a tiny pair of denim jeans lovingly crafted by Mrs Glenn. Finishing touches were suede vests, headbands and tiny cardboard guitars. The drums were made out of foil-covered cardboard and bore the words 'The Beetle-eaters'. That quartet sat on the Zoo. department desk for years and years. Brings back fond memories.
Other memories. The time an Australian Fur Seal died at Marineland. The autopsy was done in our department (poor thing died of botulism!) and we were allowed to keep the skin. Glenn did a marvellous job of curing and 'stuffing' it and Silas the Seal also graced our department museum for many years.
Another time, our department head had been travelling in the US and took with him various specimens of the 'rare' and beautiful Australian Red Kangaroo and the Euro (wallaby). In return, he got lots of great stuff for our display cases! There was a Bengal Tiger skull (which I coveted: you
should have seen the fangs on that baby!), a Kodiak Bear skull (another envy-provoking specimen) and, among other things, a Skunk's dried and preserved skin. On the day when the skunk skin arrived, I was up in the first-floor lab. I could smell it as it came in the front door on the ground floor! Whew! Needless to say, the resulting lifelike specimen was christened Pepe Le Pew and kept in a sealed perspex case!
When my third-year class did its various dissections, we were always careful to remove the skin of whichever animal we might be studying. 'Things' could be made from them! One bloke made a lovely display of his Tiger Snake's skin. Another was unable to save the skin of his Southern Snake-Necked Tortoise, so his wife made a green velvet stuffed replica and that was displayed inside the original animal's carapace (pretty nifty-looking it was, too!) I only managed to preserve my rat's skin. Sadly, the Bearded Dragon I dissected had been roadkill and so his skin was a mess. I wound up with a rather nice rat-skin hatband, though. It was a black Hooded Rat and so the skin was piebald.
The thing is that if you've ever studied biology at any level (especially anatomy), you really come to appreciate the value of a well-made specimen. The laws concerning animal welfare and the humane treatment of laboratory animals in Australia are tighter than my Mum's purse-strings, so it's only creatures that die by accident or misadventure that come the way of the taxidermist.
I have in my own possession a small collection of skeletons and skulls, one of which belonged to my cat, Figaro, who met a sticky end on the road. I prepared a goat's skull as a museum specimen for my final year exam and had to name all its structures as part of the test and prepare a pen-and-ink drawing of it from three aspects. There's no more thorough way to learn the bones of the skull, I promise.
Last story: on my twenty-fifth birthday, my then-boyfriend turned up at my door with a possum he'd found dead on the road. He told me he'd wanted to keep it for himself (he was in my Zoo 300-50 class), but in a rush of love he gave it to me as a gift. Now, how many people do YOU know who've been given roadkill for a birthday present???
I tanned the possum's hide and used it as a doiley. The skull is still on my shelf (next to Figaro and a kangaroo's mandible) and the skeleton wound up in the Zoo museum for spare parts.
In closing, I'd just like to say that people who study animals, their anatomy, physiology and behaviour, are generally immune to the 'ick' factor in the same way doctors are immune to the same thing in human beings. It's just biology, after all. I would never bother to keep a 'stuffed' specimen of a pet of mine because it's not the skin that evokes the most precious memories. They're kept in a person's heart and no specimen can touch those sorts of feelings.
I s'pose beauty really is in the eye of the beholder and if a person finds a 'stuffed' specimen beautiful, then that's his call, eh?
Now knock it off! 
