Can I Insult or Use Bad Words with a Cop?
Yes. But don't! As you already know, the First Amendment to the U.S. Constitution protects free speech. You have a right to free speech and because of this right, a police officer generally cannot arrest you, nor can the government prosecute you, simply for what you have said. However, it is never a good idea to take this idea too far and become tempted to antagonize police officers.
Police officers often rely on other charges such as resisting arrest or disorderly conduct to arrest and charge an individual. Similarly it is never a good idea to otherwise disrespect the officer (such as flipping off the officer). Although it is well established that disrespecting a police officer is not a crime. But of course, as stated, the officer can find other reasons to arrest you. Being disrespectful only makes it easier for the police officer to look for other reasons such as "disturbing the peace" to arrest you.
Also, of course if you use "fighting words" or otherwise incite violence, that is no longer insulting, it could be a crime.
The officer can find other reasons to arrest you.