In some of our really old buildings (Victorian, Edwardian) it makes more sense to insulate the underside of the roof deck using closed cell foam and then place polyisocyanuarate hard board on the topside of the deck with a piggyback roof over top.

Lay the kitchen and bath caulk on the underside of the lip, and clamp it in place with the sink clamps that come with the sink.

It rained a little yesterday and I could see the rain dripping to the underside of the fascia since the drip edge is right against it now.

The helps to equalize the temperature between the top and underside of the roof.

Basically what that means is if its inside face of stud, you are responsible for drywall and within your unit, if its outside face then you are responsible for piping in shaft that lead to unit (very rare).The other piece is vertically, which is either top of subfloor to backside of drywall (again within unit) or underside of deck to floor (which means you are responsible for piping).So a lot to uncover there along with determining where the leak is coming from, if its the drain that is you, if its within the wall, that is typically association. 

A lot of it had been fixed by a prior owner, but the underside of our house had a little of the same thing going as this one according to some prior repair pictures we saw.

Do you have underside photos, are you comfortable with the subfloor condition? 

If the standing water is a problem, you should see it on the underside of the floor-ie above where the water is. 

Me, I hope never to see the underside of a sink ever again.

In this scenario the underside of the roof deck would be ventilated to address drying concerns.