The tires are fat, with large knobs and each knob has a steel stud in it. The steel studs only help with ice and very hard packed snow, they don't do anything for regular snow. The size of the tires and the large knobs is what helps with the snow. Therefore, any mountain bike with fat, knobby tires will be useful in the snow, you'll just have to watch out for ice. Just like with riding off road, you need to keep the pressure low at around 30psi.
The Flushing Meadows Park? A.k.a. Corona park? That's a very good place to start riding, that's where I started too after a long break. Although, it gets crowded on weekends. Did you check out both sides of the park? It's really large. You can easily ride there for an hour without using the same path twice. Train there until you build up strenght and confidence to ride on the streets, then you can make connections to the Kissena Park and to the Cunningham Park on the East, to the Forest Park on the South West and to the World's Fair Marina on the North, where you can ride a bike path along the Flushing Bay all the way to the La Guardia airport. If you like airplanes, bring binoculars along

There are really no great "scenic" rides. The Flushing Park is probably as scenic as it gets unless you get out of the city. The Forest Park offers a shady, wooded path with a chunk of preserved, untouched forest.
Between these locations you'll have many hours of riding before you get bored and want more. Oh, and the Flushing park is full or birds of all sorts too

Since you're coming from 34th Ave, you must be passing very close to the Marina, perhaps there is a connection? I have to refresh my knowledge of those areas in Spring
Oh, one other thing too: you might consider joining
5BBC for their weekend rides. They have few in Queens and many of their rides are "happy face", leisurely rides. It's a great way to discover the city by bike and learn some skills, including riding on the streets. I rode with them for several years, then I learned to do my own rides. They're a friendly bunch of people and have been doing this for many years. Anyone getting into cycling in NYC should give 5BBC a shot, won't regret it. I'm thinking about doing some rides with them again this year.