I was wondering what this part of stair trim that the arrow points to is referred to as. It is like a faceplate on the side of stairs, inverse of a baseboard.
This stair trim part called
stairstrim
Related Solutions
I'd get a 2 x 4, trim it to fit the length and height of your trim piece and screw it in. 
The horizontal lines are the piece of trim that will run along the outside of the staircase.
Skirt boards are one of the hardest things in my opinion to do. I can rip apart a bathroom and have it put back together in 2-3 days and it might take me the same amount of time to make a couple of long stair skirts. All of your stairs wont be the exact same measurements, walls are not perfectly flat... One of the things that I source out now for sure.
The short answer is it depends. If the staircase will be carpeted or if there is no overhang on the treads then the answer is tread first. Basically this is faster because your treads don't have to be perfect. Also your skirt doesn't have to be perfect.
If your stairs will be showing the treads or if there is an overhang I would suggest installing the skirt and then the treads. This will give you a cleaner look. Also good luck notching out the skirt to match all of the overhangs on the tread if you decided to do it last.
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Best Answer
If it's structural (the tread/risers are supported by it), it's called a stringer.
If it's not structural, it's called skirting (or simply a skirt).