I have an up/down duplex in St. Paul, the upper unit's furnace is in the attic, and blows air down through the ceiling.  It's tucked under the corner of the south side of my roof as you can see in the photo.  This is causing a few things to happen, leading to ice build-up, drooping gutters, and water in my basement in the dead of winter.  It should be noted, this property was built in 1905, so there's a limestone foundation - water in the basement is fairly common and I can probably trace lines on the floor exactly where the water always flows.  Historically it hasn't been an issue but large snow melts exacerbate this and are causing water to pool around the base of my two electric water heaters.

Basically, the sun heats up the snow on the shingles, snow melts really fast and creates what is basically heavy rainfall down the south slope of my house, catches my gutters (which are full of snow/ice, and water pools over the side like a waterfall, coming down and creating a slushy/icy spot, causing water to freeze to the siding and create a large pocket of ice along the side of the house.  It's been warm the last few days and instead of staying ice it was all slush and the ground soaked up the water and it all went into my basement.  I've got dehumidifiers running to try and clear it up as quickly as possible.

The big question: what can I do to stop this? There's nothing else in the attic, except blow insulation, and the furnace is fully-wrapped and insulated.  A thought I had was to pay an HVAC contractor to shift the furnace to the center of the room, but it's directly over the intake, which is currently in the living room.  I'm not sure if that would even help.  Anyone with experience here?