No. They do not feel the sense like we humans do. This is because we have a central nervous system running through our vertebra (backbone) which sends messages to our brain. This means that if if our leg is hurt, we feel the pain at the top brain.
On the other hand, the insects are non-vertebra and they do not have central nervous system. Thus if an ant's leg is crushed, only the leg will feel local pain. Moreover, their pain magnitude is short, since they produce large number of off springs and their life is short. They can also reproduce their lost organs.
It has been observed that if you cut a cockroach's head, it will die after a month due to starving. Scientists have cut a lobster's leg and feed him, which he ate.
It is also true that except for human beings no other animal is aware of its existence.