I live in Buffalo. Call the Pistol Permit Office of Erie County, the county that Buffalo is in: (716) 858-6600. They're quite nice. Ask your questions about guns there. When it comes to guns: get it from the horse's mouth. Why? Because violating Gun Law in NY and many states is a major offense and ignorance of the law is no excuse.
I am sure without a NY State Pistol License you cannot keep handguns in your residence here. But ask them for official information like I said. As far as the border, Federal authorities will have the information. The US and Canada's. Ask both. At the range I shoot at, I believe Canadian gun-owners do shoot there sometimes. So, there must be some way to transport for that purpose. But you can't live here with a handgun without a permit. Call the US Customs and Immigrations Offices here. You can get the numbers from a Buffalo directory. Ask them about these questions of transport. Perhaps your gun-club could also give you informal advice. But no matter what they say: ask US Customs (and Canadian Customs about getting BACK with guns.) Seems to me at best you would only be able to bring them in for the day to shoot. BUT ASK THEM.
Getting a Pistol Permit here (Erie County) is not that difficult - unless it's Unrestricted Carry and even that is not really so hard after a year with a target/hunting permit. These categories of restrictions are imposed by some counties. Erie County does. They are county-administrative ordinances, not NY State Gun Laws. The State itself normally issues nothing but Carry Permits. The counties administrate the process of application and decision and any County restrictions they may add to the permit. In spite of its reputation, it is not hard outside of New York City to get a handgun permit in NY State.
New York City is its own world. It is close to impossible to get a permit for New York City, at least for Carry, and a special permit is needed. The normal State permit cannot be used there. FORGET NEW YORK CITY.
In the rest of the State, to apply for a State Permit you have to be a resident. And you have to have been one for a period - what it is I don't know. Ask the Permit Office.
But even if you can apply, it takes a LONG time now to get a permit. It always did - mine was 9 mos. But now it is a year or more because there are many applying - same all over the US - fear of the Obama Administration and Gun Control.
Good luck, Send me a private message if you want. But, for now: Get your info from the proper authorities. If you're on a Visa, you have a second reason for being sure of what you are doing. If you're skirting gun laws and from another country - you will have a multitude of problems and scrutiny from more Fed and Other Agencies than you want to ever know existed. Don't Screw Around! - even unintentionally! I know that. I'm on the faculty of a University here. I have foreign students regularly. I know what I am talking about. Tread Lightly about Guns!