I'm fairly certain only humans and a handful of whale species suffer from menopause or something like it. All other female animals are able to reproduce even into old age.

(Mostly in agreement with stevenjohns.) Menopause seems to be mostly restricted to some primates and whales:

> Menopause has been observed in several species of nonhuman primates,[108] including rhesus monkeys[146] and chimpanzees.[147] Menopause also has been reported in a variety of other vertebrate species including elephants,[148] short-finned pilot whales,[149] killer whales,[150] narwhals,[151] beluga whales,[151] the guppy,[152] the platyfish,[citation needed] the budgerigar,[citation needed] the laboratory rat and mouse,[citation needed] and the opossum.[citation needed] However, with the exception of the short-finned pilot whale, killer whale, narwhals, and beluga whales,[151] such examples tend to be from captive individuals, and thus they are not necessarily representative of what happens in natural populations in the wild.

> Dogs do not experience menopause; the canine estrus cycle simply becomes irregular and infrequent. Although older female dogs are not considered good candidates for breeding, offspring have been produced by older animals.[153] Similar observations have been made in cats.

https://en.wikipedia.org/wiki/Menopause#Other_animals

And is "suffer" the right word?

Tangential: where are all the dead animals?

There are plenty of animals around, even in an urban area, such as birds. But, I never recall running across dead animals, except road kill, or the dead mice and birds our cat would bring us. Is there an animal cemetery somewhere they go to bury themselves?