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.

I work in commercial construction and for balconies (which would be a similar condition) we use a traffic coating (water proofing) over the top of the concrete and a drip edge on the perimeter at the underside of the balcony to prevent water from rolling back in.

I am looking at a house from 1957 that has lead paint on the exterior that is under siding. 

A quick test for sagging joists is to put a nail on the underside of the joist on each end and then tie off a Mason line about an inch down.

Paint all wood with Killz primer to seal it, might even need to paint underside of roof.

If this were a problem with vapor barriers on the under side of the floor, could there possibly be enough moisture coming from the seemingly dry crawl space to create extreme humidity inside the house?