How do mix networks work with server downtime?
It depends on your choice of mixnet, as to whether a mixnet is resilient in the case that C doesn't participate (for whatever reason, including system failures and adversarial behavior by C). 

For onion routing, where the ciphertext is of the form 


$enc(pk_A,enc(pk_B,enc(pk_C,enc(pk_D,m))))$,


the mixnet is not resilient, because $enc(pk_C,enc(pk_D,m))$ cannot be decrypted without participation of C.

By comparison, for a re-encryption mixnet, where the ciphertext is of the form


$enc(pk,m)$,


the mixnet works, since B can skip C and send directly to D. The resulting reduction in anonymity corresponds to reducing the number of mix nodes from four to three.