tirerack.com probably has more tire info than you ever want. But here's some short answers to your questions (from a layman),
1) I don't know what the "160" means, but "225" is tire width in millimeters (mm), "70" is the aspect ratio, which gives you the tire height (distance between outside diameter and inside diameter) as a percentage of width, in your case it is 225x0.70 = 157.5 mm, "S" is the tire rating, sort of tells you how fast that tire was designed to go, "R" means radius, "16" is radius of the wheels in inches.
2) yes.
3) When you go to larger rims, you need different size tires as you suspected. You can do the math yourself, tirerack.com has a lookup tool. Idea is to keep the outside diameter of the tire the same (as old setup) but the inside diameter must of course match that of the wheels.
4) I think they make snow tires for many sizes, so that alone should not drive your rim size decision. But if you want comfort, the general consensus, I believe, is for smaller diameter wheels.
edit: under 1) "R" cannot stand for radius, one would have gigantic wheels in that case LOL, it probably stands for radial, a type of tire construction. The "16" is *diameter* in inches.