The question to be asked is the significance of residency regarding ballot access for a Presidential election.

New York has ruled that, as Robert F. Kennedy, Jr., spends most of his time in California, has most of his possessions in California, and apparently has displayed no intention of changing these particular behaviors in the near future, that his address given on nominating petitions is invalid.

Judge Christina Ryba ruled that Kennedy's claim of a New York address as his "place of residence" was a "false statement," concluding that it was evident he had no plans to move back to the Empire State and said he only listed the location for political gain. Ryba's ruling is expected to be appealed by the Aug. 15 deadline. If her ruling is upheld, it would bar Kennedy from the New York ballot and potentially spark challenges in other states where he used a New York suburban address to collect signatures. 

According to the reporting, RFKJr pays $500 per month to rent a room at the home of a family friend.

Such an arrangement would provide a valid address for receiving mail and other communication, and would be a valid address for receiving legal summons. It is no different from using a UPS store location as a “business address” when one is a consultant or other independent contractor, which is a service that UPS stores and other office service businesses explicitly advertise.

What purpose does an address serve on any of the paperwork involved in seeking access to the New York state ballot for a Presidential election that requires the address be one's actual primary residence?

As a hypothetical, how would the validity of petition signatures be affected had Kennedy used a California address instead? Or even any out of state address?

There is no state residency requirement for running for President—necessarily, as a person must stand for election separately in each of the several states. A non New York resident has the same right to appear on the ballot of a Presidential election as a New York resident.

How is Judge Ryba, by her ruling, not imposing an additional qualification for running for President aside from those enumerated in Article 2 of the Constitution? The courts have held for decades that states may not impose requirements on candidates for national elective office greater than what is given in the Constitution, yet that seems to be exactly what Judge Ryba has done.