Do all ciphers suffer from the problem of multiple equivalent decryption keys?
No. The number of non-equivalent keys is bounded by the number of permutations. Since the number of permutations is very high there is a very big chance that ciphers do not have equivalent keys. This is especially true for ciphers with a high block size (AES with 128 bits). Even if an equivalent key would exist, it would probably be rather tricky to find a set of two equivalent keys. How tricky depends on the cipher of course.
Is the existence of equivalent keys an essential property for the security of a cipher?
If you could prove that a cipher had no equivalent keys, would that lead to a procedure for recovering the key under the known-plaintext attack model?
No. Finding that a cipher does not have equivalent keys would enhance the security of the cipher, not decrease it.
More information about equivalent keys here.