@Chad Eatinger There's a lot to unpack in this thread, and I'd be happy to have a more in-depth conversation, but a few quick thoughts. I live and invest in Brooklyn, so what I say is only applicable to here though I imagine it probably applies in the more desirable areas of Queens (Astoria, LIC, Woodside, Sunnyside, Ridgewood, etc) as well.
Basically, at your budget, what you're looking for (living rent-free within 30-45 minutes of Manhattan) is not going to happen -- at least not in Brooklyn. Jersey might be a better bet, though I have no experience there.
If your budget is $1MM, I'm assuming you have around 250K in capital to work with? I'm also assuming you'll be living in 1 unit entirely with at least 2, if not 3 bedrooms. There's literally going to be nothing you can afford on that budget with 3-4 units that's within 30-45 minutes of NYU. Not going to happen.
Best case, you'd probably be able to pay a bit less monthly by buying than you would by renting a similar unit, but you'd have to decide whether that's worth the frictional costs and capital deployment for a place you're only going to live for 2 years (my guess --- it won't be). You'll also be more like 45 mins-1 hour away from NYU, if not more.
Anything on the West side of Prospect Park north of Sunset Park is going to be unaffordable.
There are 2 units in the low $1MM range in Flatbush or Bed Stuy or Bushwick (which are probably 45 mins-1 hr away from NYU on a good day) -- your monthly expense would be in the $5K range and you'd maybe get $2500 from the other unit. You can probably buy something cheaper -- 850K-900K but it would require 50-100K of work and probably be out of your budget. And ultimately you're paying more cash and still getting the same rents. It's just not really financially feasible to house-hack a duplex with a family and break even.
You can get a 3 (or maybe a 4) unit place for $1.2-$1.4MM, maybe cheaper if it's a fixer upper but again, doesn't seem to be in your price range. In those cases you can probably get to a point where you end up negatively cash flowing $1000-1500, which is not a bad "rent" for a 2 or 3 bedroom.
Other thoughts:
1. Assuming you're not an all-cash buyer, off-market, foreclosure/REO is a pipe dream, at least in Brooklyn. So forget about that. You're not going to find a way to "get in on the cheap", not here.
2. Looking for something that "needs quite a bit of work" to build sweat equity obviously requires more capital and will lower the purchase price you can get unless you're planning on going the hard money route which if you're inexperienced is a bad idea. For example, if you have 250K but you're going to need $100K to rehab the place, all of a sudden you can only really afford a $600K property.
3. L line shutdown isn't going to make properties along that line any more affordable. Owners aren't dumb, they aren't going to panic over a temporary shut down.
4. Some answers to questions --- anything 4 units and below won't be rent-stabilized unless the building has a tax abatement on it. Heat and hot water is often paid by the owner, especially in older buildings, but some newer construction will have separate boilers/water heaters. Water and common electric (hallways, etc) are paid by the owner.
Not trying to be a Debbie Downer, just trying to give you the realistic view.