drainageguttersplumbing
I'm planning on burying some rigid PVC pipe to attach guttering on my house to a few pop-up emitters like this:
But the pop-up emitter makes a right angle UP, so doesn't that mean that basically the entire buried pipe will remain filled with stagnating water? It seems like this would be normal but I'm not sure if this causes smell issues or any other adverse effects.
The down pipe from your water heater relief valve should not be plumbed into a drain. the output end may be hovering over a floor drain but not hard plumbed to drain lines. If the relief blows off due to high pressure (possibly steam) it could rupture a hard plumbed drain line.
There are two drain lines from a water softener; one over-flo from the brine tank and one off the control valve. The over-flo line could go to the same place as the heater relief output. The line from the control valve will have a significant pressure while the softener is in recharge or backwash modes and should be hard plumbed through a check valve to your drain piping to sewer or septic.
I represented a company that installed roofs and gutters in NC for 4 years. Your gutters are way too far below the shingles. Judging from the pictures this looks like an older house and you may have some issues with how level the roof line is and perhaps that is why. Ideally, you want the gutters touching the shingles on the high end and sloping down gradually to the downspout but not more than an inch below the shingles. From the pictures, it looks like the water is missing the gutters entirely when it rains hard.
You have half round gutters, which tend to act as a "ski jump" if there is any velocity behind the water flow. The way to reduce this is to use a diverter which is vertical piece of metal bent into a "L" shape with the bottom section about 3/4" wide and the vertical part about 4" wide. It is screwed to the outer lip of the gutter. I don't know if you can buy them, our guys just made them from scrap pieces of gutter. They will trap snow and debris and you will want to clean behind them from time to time. The easiest way is with an old paint brush attached to one of those painters extension poles. 
I suggest getting a well qualified gutter company involved. You have half round gutters on open rafter tails which may require changing the brackets and there are a lot of different brackets to choose from. You will want someone who knows what they are doing. I doubt you will be able to eliminate the overshooting entirely, but you should be able to greatly reduce it so that in most storms it's fine.
Best Answer
I used a post hole digger and made a small leech pit right under the 90 on mine. Filled it with gravel and drilled a 3/8" hole in the bottom of the 90. After it rains the water can drain out and absorb into the dirt.