In America everyone has a legal right to complain and most do it frequently. However, she should not pay too much attention to things written on the Internet on a site with an agenda. Perhaps she should stay off these sites.
A deposition is questioning under oath but not in the courtroom . It is informal formal questions, more than the interrogatories that you might have answered and might possibly be used at trial if different answers are given to the same questions. It also gives the attorneys an idea of the strength of their respective cases. Your attorney will prepare you for this. Your "ex" may not want to face certain questions under oath, often those are questions related to finance.
As stated, in New York you have to meet residency requirements, but other than that you ahouls be able to go ahead with a divorce. Also you would probably be entitled to some child suport. You should really discuss this with a local divorce attorney.