If there is access to underside of its floor and it's wood I would Try drilling and spraying a few cans of Great Stuff expanding foam thru the floor and see how that works. 

Adding acoustical insul between the floor joist to the under side of the floor works so.

You could drill a small hole on the underside and let any standing water drain, then when dry, you could put the cap on.

Have to insulate the attic in my duplex (frost forming on the underside of roof causing a leak). 

Are you guys talking about those planks with the fibrous underside to a thin laminate/printed surface?

In fact, there is so much air coming into the house as a whole that the furnace is constantly, constantly running - costing us a fortune just to keep it comfortable in here.I'm not sure how difficult it is to insulate the underside of the house (as well as the rest of the house).

depending on how bad it is and what you used to seal it in the first place, I would pull the drain apart and use 100% silicone on top and the underside of the drain assembly. once everything is tight wipe the extra off and you should be good as new.

Here is the crazy thing, it is leaking only when actively flushing...otherwise the underside of the tank is dry and the tank is full (not a slow leak).

I was trying to change out the faucet and couldn't really get to the underside of the sink to loosen it, I decided to remove the sink to mount it, and the sink pulled right out of the way without much effort, when I didn't even remove the mounting hardware. 

This is a closed cell type of foam that is impermeable, so can insulate a space without a poly barrier.The foam does such a good job of tightly sealing hard to reach areas and the underside of plywood where you want to avoid a cold floor.