Would a centaur be able to balance with the upper body of a man in place of the horse head?
Would it be able to survive with just a human heart or would it be better with a horse heart? How about both?
If the traditional design of the centaur is biologically impractical what changes can be made to it to make it more practical?
Assume no magic is involved.
I don't think centaurs would exist without magical or divine intervention, except in myth and dreams, for a number of reasons other answers have pointed out, such as no evolutionary path, and the fact you're combining two mostly-complete bodies, but one lacks a place to excrete (not SO bad - probably it goes into the horse-body digestive system...) and the other lacks the head (so the humanoid head would need to be able to run the horse body.
I think it might in theory be possible as the mad project of some crazy god or mad scientist or something, but it would require some sort of grafting with futuristic anti-rejection technology, incredible plastic surgery, and/or a lot of medical and/or genetic handwavium technology.
So the issues include:
What makes me think it might theoretically be possible to engineer (hey, it's more productive than figuring out how to blow up a star or make a Dyson Sphere or go power-mad at everyone else's expense), is the example of some conjoined (?) twins where there is a shared body sometimes with extra parts, and two heads.
One of my more awful ideas is there might need to be a horse brain somewhere around the inside of the lower humanoid torso... maybe not.
Maybe there's a way to screw with the genetics of horses and humans enough to make one set of chromosomes that will grow a horse body and a human torso and have them attached the way the Greeks imagined centaurs. I really doubt it, but I wouldn't utterly rule it out. Maybe it just has one brain that's someplace between human and equestrian, and that somehow figures out how everything's connected - I would not have thought some conjoined twin examples would survive, which did.
So my theoretical centaur might have a hybrid brain that handles everything, in the humanoid skull. The DNA is some engineered weird hybrid that reliably grows a functional, viable, aesthetically pleasing centaur. I imagine the growth rate would be pretty weird, since horses grow up faster than humans... there might be some large horse bodies with childish human torsos attached unless more effort is made to somehow accelerate the human growth, though the human brain takes time to learn things, so I imagine the psychology of a centaur would be horse-dominated especially before it grew up a lot, which might be about time the horse body would tend to be wearing out, unless somehow that tendency was overcome. The digestive system needs to either be horselike or human or ideally both with the ability to route appropriately - good luck with that. The nervous system should connect all to the one brain. The respiratory system is probably a huge bitch to figure out, maybe, since there's one mouth and it needs to get air for the horse body too... um, I suggest a BLOWHOLE on the back, for the horse lungs. Probably two hearts, like some human twins, and thus two circulatory systems, but hopefully compatible blood... hmm.
An alternative might be to have the humanoid organs partly or wholly un-needed, un-used, or even absent, just somehow shaped like a human (perhaps keep the lungs), but be mainly just muscle shaped more like a human torso than a horse neck, but for non-muscular purposes function more like a horse neck. So mainly you've got a horse with extra human arms and shoulders and head. Of course horses like to eat lots of hay, and human teeth aren't great with hay.
In any case, even if it is possible, I'd say it's the kind of project that would only be undertaken by people with massive resources and nothing they think is better to do with those resources.
Though maybe in the distant future there will be genetic breakthroughs that are useful for other things, which end up making centaurs not that big a deal. Certainly there is sci fi which postulates that kind of clean easy happy outcomes of genetic screwery.
The key to an evolutionary approach would be to have a niche to evolve into. Horses developed their torsos to digest large quantities of grass, and their muscular but fine limbs to run from predators. They have necks designed for eating off the ground, but giraffes and elephants eat off both ground and trees.
A route to centaurs could be to start with a six-limbed grass eater that gradually moved from ground to tree based food, such as low hanging fruit. Four limbs grow taller and sleeker to keep out of carnivore's meals, while the front two become adept at manipulating branches and feeding the mouth. This theoretically could produce similar upper limbs to humans as it is a similar functional use.
A human torso basically contains most of the organs it requires to survive, however it doesn't include the waste disposal.
The horse's body also includes most of the organs it need to survive, except, you know ... a head.
Now you're mixing the two. Which stomach do you keep? Which liver? The problems just start piling up.
So the final answer is No.
How about finding an anatomic chart of a horse showing just where the bones and organs are? Then design a centaur that is more cosmetic in the human part, showing the outward appearance of a man but relying on the horse's organs. The man-part would contain bones to support the shoulder, perhaps the spine merging smoothly together.
Note that a horse is much more massive than a man. As you point out, the total man part is similar in size to just the head. In general, keep the horse's internal organs.
In the classic sense as a creature which is a man and a horse combined together, no it's not possible.
I think this answer https://worldbuilding.stackexchange.com/a/36229/49 is onto something though. It's definitely possible to get something centaur-like but scientifically plausible.
Take the giraffe as a starting point, that's actually not too far from a centaur in terms of overall body weight distribution. Add a second shoulder and manipulating hands up near the head and you're starting to get something looking a lot like a centaur.
Grasping limbs there could be useful too so it's not implausible.
