Pic 1: If it's rubbing on the bottom, could be because you are in the small chainring and the small cog, so the chain is lower to the ground.
Solution1: Lower front derailleur to within an couple of mm to your big chainring.
Solution2: don't ride in the small/small combination of gears, you'll be fine.
If it's rubbing on the inner plate of the derailleur, you can adjust one of the limit screws to avoid this. Sheldon will help here.
Pic 2: black thingy should go over cable housing, to finish off the end nicely. You could put it over the housing shown in pic 5. Silver thingy goes on the end of the housing just where the brake housing enters your brake levers, up on the top. It helps keep the housing from getting pulled into the lever body, thus making you lose braking power.
Pic 3: All looks well in this pic.
Pic 4: The cable that comes from your lever should come straight down through the triangular metal piece (the straddle wire carrier I would call it). Can't tell from the pic, but the bolt might have a hole in it for you to pass the cable straight through, rather than merely pinching it off to one side of the nut/washer. Because it comes straight down, once you get it set up you will want to trim the cable fairly short so it doesn't hit your tire. The little silver piece (cable iend cap or tip cap) does get pinched on the end. BUT it is not there to really hold anything or take any force, just to keep the cable end from fraying. I can't quite be certain, but other than this main cable coming straight down, there should be the straddle wire, a short piece of cable that goes from one brake arm to the other, over the carrier's little groove.
Pic 5: That's a barrel adjuster. Now that you know the name, check Sheldon again.
Thanks for the pics, they help a ton.