After finally killing off all the Greek gods, Kratos went off somewhere else for a while (maybe looking to other dimensions for new challenges) just long enough for belief in him to fade. When he finally returned to this plane of existence, lack of belief had forced his power level so low he had to take on a lesser form when he incarnated on earth (much like Om in Small Gods). Even as a scorpion though, he kept some fundamental points of his personality, namely his love of REALLY violent finishing moves and his Kill Em All disposition. Him going after Struggs was an attempt to regain his notoriety, and consequently, a better form.
They've been sent back to earth in these forms to work off the bad karma the greed in their past lives accumulated and to punish Struggs for his own. They still bear some trace memories of their old lives, and consequently, their old rivalries; maybe Past!Tarantula sold out Past!Scorpion or vice versa before their deaths. As a result, each is subconsciously drawn to track down and fight the other. Meanwhile, their divinely ordained mission manifests as the Plot Direction Arrow.
In the original continuity, Spider-Man got his powers from a radioactive spider bite. And while (to my understanding) the actual comic book spider was irradiated in a lab, it's possible that (at least in Hollywood's understanding of nuclear power and radiation) fallout from a nuclear test that landed in Arizona could have done the same to a member of the local wildlife. Alternatively, the original spider could have been sent to a facility in the Southwest for futher testing after the incident with Parker, escaped, and interbred with the local population, passing its abilities onto its offspring.
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