Usually, the sump and sump-pump are located at the corner of the basement/crawl space nearest the outside corner with the best away-from-the-house drainage. How high up you run your pump out-line depends on the lift capability of the pump, but most will certainly lift the water 4 feet or so. I drilled my hole through the foundation using a impact hammer and coring drill, It should come out just above your ground line and run away from the foundation (I cover my 1 1/2" PVC pipe with mulch, its end point being about 4" below where it comes out the foundation and 48" away from the foundation. I also put a screen over the end of the pipe to keep little critters out.
The black drain pipe should extend above ground a few inches, lest it fill with dirt, leaves or other unwanted stuff. But there is nothing that says you can't build a decorative down-spout receiver to cover this pipe. I've seen some made of copper that attach to the side of your house. If you are creative, you may be able to make something.
One other thing you can do is cut the drain pipe off below ground level, make a small concrete trough (12 to 18" in diameter) around the opening, and put a screen over the opening, with the downspout dumping directly to it.