Won't matter, since Diners only provides primary CDW on a rental car that you drive. No credit card (save the American Express Platinum International Dollar Card) offers any type of third-party liability coverage. Diners won't help you here, since you caused damage to another car. (It would cover damage to your car, but you said there wasn't any.)

That's a question for whomever provides coverage on your own car. Most insurance companies will extend coverage to rental cars, but policies differ. No one here can speak to your individual situation; you'll need to talk to your insurance agent about the specifics of your policy.

The renter of the vehicle you hit will need to report it to his rental agency. They will probably ask him for information about the party that caused the damage (you). If he hasn't by then already contacted you for your details, he probably will at that time (unless the note blew off of his windshield or something).

If you want to be prepared for him, you can call your insurance company and (if they are covering you) start a claim. When he calls, you can give him the insurance company's contact information and claim number, which he will provide to Hertz. Hertz will then contact your insurance company to pursue payment.

If your insurance company pays everything, then the file will be closed. If your insurance company leaves a balance due (not uncommon at all--insurance companies rarely willingly pay administrative fees, loss of use, diminution of value, and other ancillary charges billed by the rental agency), Hertz will likely try to contact you directly to arrange payment. If you decline, then ultimately Hertz will return to its own renter to pay that balance, since he is the one Hertz had an agreement with. It'll then be his decision whether to pursue you (either on his own or with legal action) for any balance he pays to Hertz.

Oh, and I should note that the above process can stretch out into years.

I don't know how Hertz would ever find out unless the OP gives (or gave) information about the vehicle he was driving to the person he hit. It is possible that later on down the line, information about the car the OP was driving will be handed to one of the parties involved in the incident and that party may contact the registered owner of the vehicle...but I can't see any reason why that information would even be relevant, so no one is likely to push for that.

On the other hand, when you sign a rental agreement, one of the terms you typically agree to is to "immediately file and report any accident, theft, or vandalism involving the vehicle to us and the police, complete our damage/accident report, and deliver to us a legible copy of any service of process, pleading or notice of any kind relating to a claim or suit in connection with any accident involving the vehicle. Your failure to cooperate with us in any investigation or to file a timely and accurate report creates a reubuttable presumption that the incident was caused by your willlful, wanton, or reckless misconduct and, as a result, may void any optional products you purchased and make you absolutely liable for any loss or damage to the vehicle." (Quoted from one rental agency's contract verbiage I just happened to have handy.) I have seen some insurance companies determine that failure to comply fully with the terms as outlaid in the rental agreement is sufficient grounds to deny coverage, so there is a risk of that in this case, however small it may be.