You need to define what the word alive means first.
If you cut a roach’s head, it stays sort of alive in that the body responds when trying to touch it or expose it to harm but it won’t do anything if left alone. just standing still until eventual death.
A decapitated roach might still be able to “feel” pain but in my personal opinion - I am no expert in the matter - it doesn’t feel stress, trauma or have bad memory of the incident. Just pain and an appropriate reflex to it.
No it hasn’t been. Cockroaches, lobsters, and other arthropods have a decentralized nervous system. They have ganglia throughout their entire body instead of one centralized brain. These ganglia can react to stimuli independent of the other ones.