Wire identification needed

That looks like the vent line for the EVAP vapor canister if you mean the circled hose. Unless it is leaking I wouldn't bother with it yet. If however it is leaking you can get a new one from places like Rock auto online. They are about $200.00 or so.

I can't blow up the picture on my end, but I think you are talking about the wire that connects to the canister vent solenoid. It's responsible for venting the evaporative emissions system when the ECU commands it. You can either buy a replacement pigtail new or at the salvage yard, or pull the pins out of the connector and heat-shrink each wire and put it back together, but don't use regular electrical tape because it will come off over time.

No, the hose and box is part of the system that removes the fuel vapors from the gas tank. When you fill up or the vehicle sets the fuel vapors are collected in the charcoal inside the box. Then at certain times the PCM will energize the purge solenoid and route the vapor into the intake system to be burnt. If the hose is leaking or has gotten bad from fuel vapor it isn't hard to replace, you just need a section of fuel safe hose to connect the two pieces just like the one that is there. Unless the connection is broken off or it is leaking fuel vapors it's still okay. It wouldn't cause anything brake related though.
For that see if you can shoot a video and upload it with a new question about the brakes.