In Harry Potter we learn that Voldemort has split his soul via the Horcruxes and Slughorn mentions it as if it is a negative thing.
What are the actual problems caused by splitting one’s soul? Is this why Voldemort was disfigured?
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Splitting the soul has only two canonical, confirmed effects:
1) It will allow the person to survive (in a fashion) the physical death of their body. The do not quite die, but they are no longer alive. The residual wraith-state allows them to posses others and (given the proper magics and support) eventually regain life.
2) It makes the soul easier to fracture. Eventually, it will cause the users soul to be so damaged that they can unintentionally cast out their soul when using powerful magics.
These are the ONLY effects seen in the books and confirmed within them.
While a horcrux exists, the creator cannot be completely killed. It functions like a limited phylactery does for the lich in D&D.
The other effect we've seen, where Volemort unintentionally implanted part of his soul into Harry Potter, was a function of his soul (already torn apart many times, and missing many pieces) being shattered by the rebounding magic and a piece latching onto Harry.
This would not be a common effect.
As for the rest of your question, we cannot assume Voldemort's appearance is a result of creating Horcruxes. He is repeatedly stated to have undergone many secret, Dark rituals. Any or all of them may have contributed to his appearance. The actual cause of it is never stated.
That said, mythology has typically attributed the lack of a soul with increasing deformities in formerly-human monsters. It would be totally appropriate, thematically, if the continual loss of his soul (and, therefore, humanity) had outward physical effects on Voldemort's appearance.