We didn't "evolve" from the other primates in the sense you're thinking of. We are primates and share a common ancestor way back when, but our actual evolutionary path became seperate from all the other living primates millennia ago. We didn't evolve from Chimps, we're related to them. It's this relationship though that gives us very similar bodies and brain structures that would allow them to evolve to fill our environmental niche should we somehow vanish from the Earth.
As for insects, if you're talking about simple numbers and survivability a vast majority of insect species will survive the same events as cockroaches and have done for eons. In terms of numbers and ability to survive insects have ruled the earth over humans throughout all of humanity's history. But we're talking about creatures that are likely to develop human-esque intelligence here from what I can understand. Insects and the like have tiny little bodies that can only hold tiny little brains. To even begins down a path like ours they would need to get hundreds of times bigger as well as develop significantly more complex brains and nervous systems. Not to mention they'd also need to develop a viable substitute for the human hand. Basically, if you're talking survivability, bugs already rule the Earth but if you're talking filling a similar niche to humans, they never will. There's just too many animals that are far closer to that level for bugs of any sort to ever reach it.
