It is not too late to file a claim however it doesn't help that not report was filed on the date of the accident. The mere fact that an accident occurred does not mean that someone is liable for it. Was there something wrong with the escalator. Your best bet is to consult with an attorney in your area because there may be more than one entity involved that should be put on notice and to do so as soon as possible.
As my colleagues indicate, no claim exists against the City absent a special relationship. Your real claim is against the bicyclist. Not intending to be glib, but you'll have to write your local congressional representative if you feel the rules are inadequate.
Your attorney is in the best position to answer this question. That being said, it is unlikely that the insurance company would want to take the officer's testimony unless they deemed it favorable to the defense. You're right, the officer probably won't remember the particulars of your accident but if his report is favorable to the defense, they may want him to simply confirm that fact. Again, you should really discuss this with your attorney.
New Jersey has a 2 year statute so it sounds like, yes, someone can bring the action. Unfortunately it is not unusual for people to wait until the statute is expiring to file their action. Turn the suit papers over to your insurance company right away.
As Mr. Gennarelli indicates, your entitlement to a portion of the settlement proceeds will be a function of whther or not there is a will and its terms. You should be able to get a copy of the wrongful death compromise petition which will state who gets what and how much. If it appears you are getting nothing or what you believe is not the proper share, consult with an attorney before the compromise order is signed.
You should absolutely file a workers compensation claim. As my colleagues point out, you may have a third claim against the manufacturer of the coffee machine. It seems a little early to determine if your scarring will be permanent given the newness of the injury however it is certainly worth consulting with a personal injury attorney as soon as possible.
As my colleagues have indicated, the answer is yes. Post no more information here and put all carriers on notice of a potential claim. You may want to speak to an attorney BEFORE making any statements to any carriers however.