I believe that cockroaches (nice name, when you think of it - but I disgress…) are evolutionary purposeful where they originally evolved. They fitted right in in an environement they had magnificently adapted to: they reproduced just fine, only to be eaten by the right predators at just the right rythm. They did so for 300 - 350 million years, branching out into 4,600 species (of which only 30 are considered pests). But then humanity came along and gave them a niche to become a most succesful invasive species, and they used it well (so far).
It is us who should not have taken them out of their original habitats. Don’t blame them for making the best out of our misguided actions.
Spare a kind thought for them: when we disappear from this planet they (the pests, anyway, 30 out of 4,600) will find themselves in a lot of places they cannot survive any longer without our central heating and the garbage we leave lying around. I am almost as sad for them as for us.
And without them we would not have the song la cucaracha.
Do humans?
Though these species are hated by many, but, still can be a proper food for some animals; the animals that prey regularly on cockroaches are:
Toads and frogs
Lizards
Certain large species of beetles
Certain kinds of parasitoid wasps
Entomopathogenic fungi
Which begs the question, does Entomopathogenic fungi have any positive, useful purposes at all? 
Wasn’t there a bit in a Discworld book about the creation of cockroaches.
Yup. They help keep insect populations in balance.
(And they presumably make money for all the people selling them.)
Everything eats something else. And everything needs to be eaten by something else. That’s how the system works. When a living thing moves or gets moved to someplace where nothing eats it, it’ll do damage until something eventually brings it back into balance.
(Mandatory mention of the Smothers Brothers comedy classic “Pancho Villa Without A Shirt”).
La cucaracha, la cucaracha,
Marijuana que fumar
That reminds me: one time I met an entomology student who made a few bucks breeding and selling pet cockroaches.
…”So I bought a Roach Motel to help with the problem. At first, I was excited, because I had never been a property owner before…”
Little known fact - they were the one critter that survived the flood that was NOT on the Ark.
Some entertainment value:
I worked for awhile as a cook at a restaurant in downtown Detroit while putting myself through college. I hope this doesn’t come as a surprise to anybody, but every restaurant in a big city has cockroaches, no matter how fancy and scrupulously clean it’s kept. The owner of this restaurant was very scrupulous about cleanliness and cockroach amelioration.
I also did some cleanup, mopping and such, and the owner was going over with me the importance of making sure everything is thoroughly and completely cleaned up, even something as seemingly innocuous as a bit of spilled drink in a corner. “a cockroach can survive for a year on a single spilled drop of beer” he said.
I replied “yeah, but it would probably go on a bender and finish the whole thing in one night”.
Perhaps cockroaches will be evolve into the species that finally colonises the universe and become so powerful they fight entropy itself and fill the stars with life, joy, happiness, and meaning.
From the order Dictyoptera 
I was of the impression that even a nuclear weapon wouldn’t kill them.
To exterminate a human: 10 Gy
To exterminate a roach: 1000 Gy
False eyelashes.
Nitpick: from the order Blattodea, superorder Dictyoptera.
Nits, BTW, are the egg cases of head lice, from the family Pediculidae. Before somebody asks, they are good for picking, evidently 
.
Or maybe a neighboring unit is infested. Like a neighbor’s kitchen so covered in grease that it’s basically one big roach bait.
They taste like chicken. Perhaps.
AIUI, the resistance to radiation thing is a bit of a myth.
In general the field strength an organism can tolerate is strongly inversely correlated to volume, for obvious reasons.
Cockroaches being so much smaller than us can tolerate much stronger fields, but they are beaten by many slimmer bugs.
Of course they are a highly adaptable species and can live in many kinds of shelter, so it’s hard to imagine a scenario where they die out without planet earth being essentially sterilized.
However the same is pretty true of humans. Love us or hate is, there are millions of us under every rock, and we’re adaptable as fuck. It’s not gonna be anywhere near as easy to eradicate us as movies imply.