Cycling in NYC is pretty safe, especially compared to other big cities like London and Paris. If you are a confident rider you will be fine. Commuting from anywhere in Brooklyn and Queens is doable (assuming you are working in Manhattan), but could be long if you are at the far flung reaches of each borough. The only borough I hate cycling in is the Bronx, the car drivers up there seem a touch more hostile. All of the ‘good’ neighborhoods in Brooklyn/Queens are commutable by bike daily (e.g. LIC, Astoria, Dumbo, Park Slope, Williamsburg, Brooklyn Heights, etc, etc). I used to do a reverse commute (Manhattan to work in Prospect Lefferts Garden (not a particularly good hood)). There are definitely roads to avoid (e.g. Northern Blvd in Queens, Atlantic in Brooklyn) but there are cycle routes in both boroughs that will get you where you need to be.
Yeah, I don’t think NJ is an option unless you stay in Fort Lee, Englewood Cliffs, etc. The only way in to Manhattan by bike (and riding it) is via the GW Bridge. If you are working in lower/Midtown Manhattan, then this will be 10+ miles just from the bridge (NJ side). Ride is fine though and the most popular cycling route on a weekend. Can’t talk about riding through the local NJ areas other than the areas north of Englewood are very nice (e.g. Tenafly, Demarest, Alpine, etc). You could live in Hoboken, Jersey City etc, and take the PATH or Ferry in to Manhattan and then make use of the Citi Bike system (best thing that has happened for transit in NYC…I almost never take the subway now). You can even do Citi Bike from certain places in Queens and Brooklyn into Manhattan.
Connecticut…what!!! That is a long commute.
Long Island…also a long commute. You are looking at about ~20 miles just to hit the border of Nassau County, definitely doable every now and then but I wouldn’t want to make it everyday. Remember that 20 miles will take 1.5 hours or so cos of traffic lights, traffic, shit roads, and so forth. I have ridden twice in the last couple of weeks out to Nassau (north and south i.e. through Queens and Brooklyn) and I definitely wouldn’t want to do it everyday, although there is the Brooklyn-Queens Greenway, which is very nice.
Finally, if you can, I would live in Manhattan!! Personally, I like the Villages and the East Village/Alphabet City still have some ‘bargain’ rents around. LES is good and somewhat affordable as well. The other nice areas, such as Greenwich Village, West Village, Chelsea, Soho, Tribeca are all pretty expensive (or really shit apartments) but all very nice. Battery Park isn’t too bad. Financial District has some good offers but the hood is crap, dies at night. I have a ‘thing’ of trying to never go above 14th street unless I am trying to get somewhere, so any neighborhoods above there I think are pretty awful (yes, that includes the UES, UWS, although benefit of both is Central Park but way too many families/strollers big stores, etc, for my liking). Midtown is tourist central.