Joe, coop's are unlike condo's, private housing or rentals. I've been a board member for over 20 years, and I am not a lawyer. Your father-in-law purchased the right to live in his apartment. There is no given right to succession, regardless of length or time or relationship, unless spelled out in the lease and bylaws of the coop. In my coop, it is not uncommon for an owner to request the addition of a spouse to be added to the lease.
We check out the spouse as we would a buyer, to see if he/she appears to have the financial resources to maintain the apartment in the event of death of the owner. Generally, we approve those and issue new share certificates/leases. We have had similar situations regarding adult children, generally not resident. So far, no one has complied with our documentation requirement in order to have another name added to the shares/lease. This is generally just a move to attempt to simplify estate proceedings. If it is not likely that the party would actually move into the apartment, we generally would not permit the name to be added.
In your case, your wife has been living there for a lengthy period of time, if she has shown finances capable of maintaining the apartment and paying maintenance, and has a financial reserve to fall back on in the case of an assessment, the board should be willing to add her to the shares/lease.
You can do a little research on the NYC website, some coop's record the shares/leases and you could check the building records to see if others have had additional names added to shares/leases.
Have you or your wife spoken or corresponded with anyone on the board? Or asked to meet with them? The management company/agent doesn't make decisions, the board does. The coop board may not want any lawsuits relating to your dog, so if you can resolve the issue with the vet bills, you might have a better chance of getting approval. You might need to agree to give up the dog as well if they fear a repeat. You can request or prepare a 'stipulation of release' (or have a lawyer do it) releasing you from all past and future claims for damages and claims from the one incident with the dogs, prior to turning over a check. Good luck!