No you can not live together if you have a legal seperation in New York state. To do so makes the seperation null and void and in order to begin the divorce process they will have to get another legal seperation before they can file for divorce. In a legal separation nothing much changes other than the two parties live seperatly. They will have to discuss with their lawyers how they want the finances to be divided and whether they maintain the same insurance etc. It depends upon wht they agree upon. Sometimes an arbitrator is needed sometimes it is necessary to go to court. But first they have to decide once and for all just what it is they want, they can't keep going back and forth because they are wasting the court's time and money by doing so and that is illegal.
Any agreement made over $500 should always be made in writing. However, that doesn't mean verbal agreements are completely null. I want to say your in laws are screwed, but the only option I can think of is get an attorney's advise or try Small Claims Court. If you have proof of work done in the property, i.e. hiring a contractor to fix the roof, and you have a receipt, an affidavit from the contractor, etc., you may be able to convince the judge and get more than 4,000 back. But you will not get the house nor can you make the couple add the names to the title.