Identity Theft: Counterfeit Checks but no Traceable Bank Account

I’ve been dealing with an ongoing identity theft problem for the past 4 months. Back in March, somebody stole my wallet at a restaurant and it was returned to me by the staff completely empty. All I had in there was 2 bank cards (1 credit and 1 debit), an expired NY learner’s permit, and a school ID. I promptly cancelled the credit/debit cards and had new ones sent to me.

Just recently, I started receiving mail from debt collectors asking for payment corresponding to several clearly counterfeit checks written in my name and using my address. The letters included the store where each check was submitted and some even included a photocopy of the bad check. I’ve been calling each store, and at one of them, the store clerk took a note of the fraudster’s ID (because he also thought something weird was going on). The ID he wrote was exactly the ID on my NY learner’s permit so I assume that the fraudsters are the same people who stole my wallet back in March.

The weird thing is that these routing and account numbers seem to make no sense. First of all, they are not linked to my own finances, and I have not seen anything happening in my accounts. These fraudsters seem to be making checks with fictitious account numbers and routing numbers taken from random banks in the area. I’m extremely perplexed as to how these bad checks are even getting cashed in by the stores they’re going to, but I assume it’s because the store clerks may not be able to tell a fake check from a real one. Obviously, the debt collector letters are extremely frustrating, and I have filed a report with my local police so that I have proof of fraud. As per instructions I have found online, I am going to reply to each debt collector with a letter explaining this identity theft and a copy of the police report. From what I understand, this should not only absolve me of paying the fraudulent charges, but also ensure that my credit is unaffected.

I haven’t found much information here or elsewhere about this specific kind of situation where the fraudster is just using my name to cash counterfeit checks (i.e. no ties to my finances). I was wondering a few things:

(1) Is my understanding of how to handle the situation correct?

(2) As these fraudsters continue writing bad checks, am I required to continuously update my police report and contact each debt collector one-by-one to keep ensuring that I am not liable for these fraudulent charges?

(3) How is it possible that someone can print and use fake checks in this manner using just my name and address??? Should I be worried that there could be some bank account in my name that I don’t know about?