Where are all the dead cockroaches?
In the big cities of North America (e.g N.Y), the big piles of dirt and moisture attract many hostile animals such as the ones you mentioned: the cockroaches. You should also consider that in the north there is a very common factor that attracts collonies of cockroaches: the low to mediocre temperatures. Moreover, you should reconsider the cleanliness of the sewers. The sewer and the leftovers that exist there are highly nutritional for the cockroaches. That's why you don't see many dead ones on the streets. And where is the start of the sewers? The sewers start from every house. That's why you notice more roaches in the houses than outdoors. They need a place that might provide them the sewer's environment such as the bathrooms, or an other place in the house that has moisture and is mostly dark. Cockroaches live and die to the same place and don't go somewhere else to die. The decay rate outdoors is higher of course, than indoors because every house is cleaner than the roads. With the process of cleaning many factors, such as mites are swept away to the environment.

For the second part of your question:

Outdoors there are many  scavengers and natural predators for the cockroaches. Some of them are: toads, frogs, beetles, geckos, iguanas and birds.  Entomopathogenic fungi have also been identified as lethal to roaches, as their spores attach to insects and kill them within weeks. Cockroaches affected by fungi then pass the parasite on to other cockroaches, and the cycle continues. 

Ants prey on some cockroach species, but other roachess actually live with ants in their colonies. These are called myrmecophiles, meaning lovers of ants. They don't seem to offer their hosts anything in return for living with them, and are usually tiny and unobtrusive. Other species - termitophiles - live in similar fashion with termites.

And yes ants are the major factor of dicomposion in nature, because they devour almost anything.

I hope this answered your question, becaused I used very extended research :)