So, let's tackle this one by one...
The black box on right side of bike with full and low - this is your coolant overflow reservoir. It is designed to hold an amount of fluid (between those marks) and is connected to the thermostat/radiator cap. When the bike heats up, excess coolant gets pushed into that bottle. When the bike cools down, it sucks fluid from the bottle to fill the coolant system.
There are two hoses that can come off there. One is on the back side of the bottle, at the bottom, on the left hand side. It should go up to the right side of the gas tank by the neck of the bike, where you fill the coolant. The other hose? On my bike, the end sits under my leg and burns me when I have too much coolant in my system and it boils out.
The tubes from your gas tank by the seat are part of the [I]gasoline vapor control system[/]. The right side tube does something fancy with your California emissions system (unless it's a California tank on a non-California bike, in which case, Block it off.)
The left hose controls gas tank pressure, and just needs to vent to atmosphere. Nothing fancy.
I'd like to see a picture of the device in front of the radiator. If it's a black plastic box, that looks like it goes into the frame, it's part of your air intake system. Someone else can help you better here.