It seems to be leaking something. I noticed about 4 hoses that need replacing.
Does anyone know what this hose is called? I need to order it.

There is a yellow/plastic box next to this hose as well. It has its own, separate hose that also seems to be leaking. What is that plastic box?

 

Looks like its power steering.

Does the hose that the arrow is pointing at a power steering hose?

I know it has another, thinner, hose coming out of the power steering piece. But is the bigger hose (the one being pointed at) also have something to do with power steering?

 

The one you are pointing at is a power steering hose

 

x3 power steering.
The plastic box is probably the power steering reservoir which is basically the back half of the power steering pump. There is a hose that comes off the bottom of the reservoir that goes down to the power steering cooler on the frame rail under the engine. All the "cooler" is, is just a metal line that loops in a circle a few times along that frame rail. Chances are one, or both of those lines are leaking. From the factory, the return line is crimped on to the cooler. However you can cut the crimp off (or just cut the line below the crimp) and use a piece of rubber power steering line with a hose clamp.

Make sure you use the new o-ring that comes with the new line, and it would be a good time to flush the system out as well.

 

I devised a cool way to flush my PS system. I had a friend pour the reservoir full of fluid, then undo the return hose into a bucket. Then I used a long belt looped around the PS pulley, and the top looped around a hand drill with a large hole saw (or other round cylinder you have in the garage). Then turned the pulley at full speed with the drill. It purged the PS fluid without having to crank the engine.

 

My power steering hose looked similar on mine with the wet mark, but my fluid level does not move in the reservoir. I haven't added any fluid so I am leaving it alone. My guess is when pressure builds up the seals hold well and when no pressure there is not enough pressure to leak out. I would want to make sure it was leaking first and it hasn't happened yet.