Keiko,
I don't think anyone who is a regular contributor to this site is a NY Matrimonial lawyer.
Your question is specific to New York. Seek the advice of a NY atty - many will offer free consultations.
Cohabitation makes a difference in many states - in CA, it could terminate the divorce process and can be the basis for recalculation of child support arrears, and non-payment of alimony (monies already paid, however, can be seen as gifts or maintenance).
Even if the parties decide to continue the divorce process, the original petition, as filed, is now incorrect. The stated separation date has changed.
The date to divide debts/assets is murky.
The correct process is for the Petitioner to dismiss the action, advising the court that the parties are now reconciling, and ask the court to terminate all orders previously made.
If the parties remain together and allow the orders to remain in place, the obligor should keep proof of the cohabitation (names & addresses on file with school [update them], mail delivery, addresses for work, driver's licenses, insurance, etc.)