Cheating death and fighting communism: that is how a fellow officer once described our job.  It was meant to be funny, but as time went on it seemed all too true.
I spent more than ten years in law enforcement, all of it on the street in uniform patrol.  I've been a patrol officer, instructor, sergeant and lieutenant. 
Do not report crimes here. Nothing here should be considered legal advice. All opinions are my own.
No exact routes are ever taken/repeated. It would be a good way to be ambushed and that's never a good day.
                             Every department has their own protocol.  First duty is to protect life:  citizens, officers, criminals (and in that order.)  Until the scene is made safe, no investigation can begin.  Once everything is safe, medical aid is rendered for anybody that is injured.  By this time, a sergeant or lieutenant is on scene and takes command.  Depending on the agency, either they or an outside department will handle the investigation of the use of force.  A CSI unit would probably be involved, but they are only a support unit to document and collect evidence.  They don't do any investigation.
The involved officers give a brief statement to the responding supervisor about what happened and are separated from the other officers who are investigating the incident.  How/when the officers are interviewed varies from department to department.  Officers are entitled to legal counsel as is anyone being investigated for potential criminal acts.
  
                          
                             No, stopping someone without reasonable articulable suspicion would be a violation of the 4th Amendment of the US Constitution and a bunch of state laws and department policies.  
Speeding, even if "everyone does it" is still illegal, hence the reason why you were stopped.   
                          
Call your local police department and ask for their assistance. They will know what to do.
TV Meteorologist
Is having a weatherman really better than just throwing to a 5-day forecast screen? 
                                        
                                        
                                             Hollywood Executive Assistant
Are Hollywood execs as nightmarish as depicted on TV? 
                                        
                                        
                                             Toll Collector
Do you think there will be a time where all tolls are automated? 
                                        
                              Reasonable suspicion and probable cause are built by the totality of the circumstances, which may include a citizen complaint. It is impossible for me to tell you how the officers in your case established either since I was not there and I am not privy to the details of the investigation.
                             Why would you think that it would be even remotely ok to shoot children with a BB gun?
1.  None of this is legal advice.  Consult an attorney.  Wrongful use force against another, especially a child, can land you in prison for a long time. 
2.  The use of force is generally resricted to "reasonable" self defense.  Re-read #1.
3.  BB guns are not toys.  They can and have killed people.  Use Google and you will see multiple stories about BB guns killing people.  For example:  http://www.foxnews.com/us/2013/03/21/florida-boy-10-dies-after-brother-accidentally-shoots-him-with-bb-gun/
4.  BB guns can also take out an eye or do other serious injury even if they were incapable of killing.  Re-read #1.
5.  You're not Alvin York and your aim on a moving target isn't that good.  I don't care how good a shot you think you are, you can't guarantee where you would hit someone anyway.  Re-read #1.
6.  None of this excuses whatever trespass these children may have done, but shooting them with a BB gun sounds way beyond the realm of reasonableness.  What would you do if your child had been shot by an adult with a BB gun?  Re-read #1.
7.  Use of force is generally restricted to the defense of persons, not property.  Re-read #1.
8.  If someone is coming into your home and you feel threatened, things are different.  Re-read #1.
9.  Re-read #1. 
                          
                             I'm sorry, but I'm not clear on the description of the intersection.  My previous answer is based on a standard cross-type intersection.  I'd have to see the intersection and know the laws of your state to give you my opinion (which has no bearing on anything, mind you - for that, contact an attorney.)
But, if you enter the intersection on anything other than a green, then you are likely violating the laws of your state.  If you enter on a green, it is going to depend on the circumstances and how the laws are written. 
                          
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