Fuel vent dripping fuel

That "Box" is you climate control/ blower housing. Yes coolant does go into it. Those hoses go to a Heater Core which may have sprung a leak, or the hoses to it have failed. Yes you will need a few tools to get that out and open for repairs.

Sounds like a blocked breather. Be careful with all that leaked petrol.

Hello and Welcome to FixYa!
Well, it seems like the drain pipe or the drain channel is clogged, and the defrost water from the evaporator coils is dripping through the quoted regions, down to the base. Try checking the drain hose/ channel and remove the clogging if any (Blow some moderate temperature air through the channel and check/ remove any permanent clogging If any). Feel free to ask the details, or any question relevant to the trouble..
Concerned.

There should NEVER be fuel from the tank even when topping off, only a burp from the filler neck if any.

There are a few hoses to the EVAP and Vent system over the tank that crack and will leak. You really should have it inspected soonest.

This saw has a fuel tank vent that could be leaking a small amount of fuel, or the fuel line coming out of the tank could be leaking. Fuel line replacement isn't complicated, you could try installing a new one. If it's the vent, odds are it's not enough to cause alarm........unless the vent line is cracked. In that event, you should replace the vent hose. Again, not a complicated repair.

Your bike has a vacuum operated petcock. The vacuum diaphram inside the cover set has cracked and is allowing fuel to leak from the vent hole. Unfortunately, you cannot buy just the diaphram, you've got to buy the cover set (#5 in the picture below), P/N 16953-KEA-023, or replace the whole petcock assembly (item #2), P/N 16950-MEM-674.

You may have a steel line rusted to the point of seeping gas back there also or a hose getting porous with age.

hello,seems you have a loose clamp in fuel hose or your fuel hose have crack in area of clamp,also you can visual check fuel supply line,fuel filter up to primary pump,ty

I had the same problem and I traced it to the fue line leaking. On the back of your unit just inside the cabinet where the fuel line is attached is a valve. I suggest you slide a piece of dry paper towel under the valve to see if after time you get a fuel stain. If you do get a stain here, you need to tighten the connection on the supply side of this valve.

Check your hose connections in the back for tightness and sealing (there should be some rubber seals inside the screw-in connections) and if tight and not leaking, you might have a loose clamp inside the unit, or a leaking/damaged hose, or even a defective/leaking pump. If it is brand new and returnable I'd return it for a new one rather than try to fix it...