After he had killed her he left her body to putrefy in the cellar.

yes because it can decay the food

Decomposition , molder or putrefy ; see link to additional synonyms .

Any dead creature in the water will putrefy and the resultant putrefaction will kill any fish in that water. So yes the dead crab will kill the Betta eventually.

Generally no, unless blood supply is returned to the area or a supporating bacteria somehow invades the area. One of the more common locations for an infarct is in the kidneys, and the usual chronic result is fibrosis, not putrefaction.