Definitely not! Cockroach eggs are contained in a case called an ootheca and if this is damaged the eggs inside will die. Some cockroaches deposit the ootheca soon after it has hardened, a few carry it until it hatches, and others incubate it inside their bodies until the eggs hatch.
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Yes, they are considerably more resistant to the harmful effects of radiation than humans and other mammals. However, there are many insect species which are much more resistant to radiation than cockroaches. Click here for more information.
Probably because after they die the muscles in their legs contract due to rigor mortis, pulling their legs towards the body and flipping them onto their backs.