1. CIA "agents"
It's a common mistake, not only in Hollywood. Media do it all the time. The correct term is "officials". People recruited for espionage are agents. FBI agents, NCIS agents. We are officers.
2. Their operations end up with high speed pursuits and shootings
If they do their job well, no one will know that they were there. Unless it's a war zone, they don't need to carry a weapon.
3. They arrest the bad guys
CIA has no authority to arrest. They can provide informatio to the FBI or another agency, and then those put the suspects under arrest.
4. Everyone at CIA is a spy
Operation officials recruit agents to steal secret and espionage. Around them, many personnel operators, security analysts, and officers. Not everyone is a spy.
5. No one can know you work at CIA
You won't go running and yelling your job position, but it's not true that telling is prohibited under death penalty or permanent or life imprisonment.