Avoiding an emergency vehicle absolutely justifies disobeying the sign. However, don't expect the police officer to agree that there was an emergency vehicle forcing you to do what you did. Otherwise, why did he write the ticket? There are other things to look at in a case like this too. Color of car, unfortunately isn't one of them.
Sounds like you may be talking about a red light camera ticket? I'm not sure. If you are, you'll have to wait and see if a ticket comes in the mail. Good news is camera tickets do not go on your license at all.
If you'd like to sort it out yourself, you'll need to contact the particular court or agency involved. If it was Brooklyn, it's the TVB (you can try to find the case information here: http://www.dmv.ny.gov/PleadandPay/default.html)
If you can't find what you need or would like some help, you can always give us a call.
If you decide to have a trial on the matter, it can be harder for an officer to properly answer questions on cross exam if he didn't witness the incident. However, it's not impossible--in NY, an example might be an officer who testifies that skid marks were present after a rear end accident on the highway may be able to successfully sustain a charge of following too close.
We do TVB appeals. It's hard to tell exactly what happened from your description. Regardless, the transcript of the hearing will be the most accurate representation of everything that was said in the courtroom. Prior to reading the transcript, it's impossible to judge your chances of success. Generally, TVB appeals are quite difficult to win. You have 30 days from the date of conviction to file the appeal. Contact us if you'd like to discuss.
You're not anything until there's a final decision on the case. You did two things wrong here. First, you entered the intersection without absolute certainty you could make it to the other side. That is the driver's responsibility, not the vehicle in front of him (assuming the vehicle in front hasn't just stopped for no reason in the middle of the intersection). Second, you honked your horn at an officer. That's never going to help a situation. You can fight it like anything else....
I would expect to hear about this one way or the other. I'm sure the officer wasn't too pleased. What may happen next is at least partially dependent on what you were pulled over for in the first place and may be exacerbated by your leaving the scene (additional failure to comply w/ a lawful order charge).
I would certainly clear up anything that may be an issue. I've seen open and timely matters get in the way of immigration applications so certainly delinquencies/suspensions/warrants can be an issue. I would also consult with an immigration lawyer...