I'm impressed!
1. Hydrogen is the right answer indeed and of course was used in the original Zeppelins.
2. to give you a bit of the information I withholded: New York is about 50000 sqare kilometers. Even my tiny hometown Vienna is about 500 sqkm.
3. Yes I'd go with 1l/m^2 = 1mm rainfall for noticeable or with anything in the dimension of 10^0 for the sake of easier calculation.
4. We are of course disregarding that the explosion would be huge and have some effects that would prohibit our rainfall from occuring plain and right away.
5. Okay here comes my hastily put together calc.
So assuming I wanted 1l of rainfall in NYC
50 000 sqkm = 5*10^10sqm => we need 5*10^10l = 5*10^10kg water
1 mol H2O has 18g therefore we need
5*10^10/18*10^-3 mol = 2.7period*10^12 mol water
thus 5.5period*10^12 mol hydrogen
now we know that under normal pressure a mol of any gas has about 23l
that would make 1.27per*10^14l = 1.27per*10^2ckm
1.27per*10^2 = 4/3pi*r^3
3.05*10^1 = r^3
3.125 = r
We'd need a balloon with a diameter of 6.25km for NYC.
If you make the same calc for 100sqkm the solution is about 787m diameter so right there with your half mile.