Question: How can Murphy still be alive if he doesn't have a heart?
Answer: Watch again the scene (s), especially close to the end when his armor is reverted to silver and just between his lungs; his heart is just behind his lungs and beating, though if you blink you might miss it.
Question: Many times, when Robocop's helmet screen is not in front of his eyes, he can still get and see information about people. How come?
Answer: The helmet doesn't contain a screen - it's purely there as protective armour. Murphy's heads-up display is built directly into his eyes.
Chosen answer: He most likely has an artificial heart or some type of system that circulates his blood, this system doesn't necessarily have to take the shape of a typical human heart. However if his biological heart is still functional, it may be behind the lungs.