Sounds like that is a GFCI outlet, so it is very odd that it is still on (and green?) when the outlet doesn't work. Yes I would think that turning off it's breaker would shut off the light.
Where I've had water leaks in the 30 years I've lived here: When I first bought the house I had a regular leak along the opening where the addition met the main house. Two roofers did the "add tar" thing which didn't work. Rerouting the upper gutter drain all the way to the ground helped that, but then the new gutters were improperly hung, the new roof didn't have the flashing or drip edge(?) to keep backed up, overflowing water from going under the shingles - and then the gutter covers themselves becoming clogged with a layer of maple flowers & maple seeds (spring), or leaves (fall) caused water to get in. So now I know that I have to clean them off at minimum twice each year.
I've also had a leak from failed sealant around the plumbing vent pipe where it exits the roof, and from a popped nail in a single shingle, and most recently this past year from absent and inadequate caulk and flashing in areas after my siding and roof were replaced (damage from a tree falling on the house).
Even in this large city, it can be so hard to get competent, ethical workmen. I've learned that neighbor recommendations have to be taken with several grains of salt.
Q