#1) Either way, your choice. It maybe hard to match your flooring exactly if using maple plywood. Another option is paint grade wood. You can use MDF, poplar, paint grade plywood, etc. and no concern about matching the flooring.
#2) It can be done this way. It is sorta like cabinet refacing as you are veneering over that rough framing lumber. But in the correct sequence of events, one would install the stringer/skirt veneers first before installing the treads. This way, you only need to be about 1/2" within your measurements so it's much easier. The flooring will but cut net and tight and will cover any gaps. You could also use paint grade stock as well. Or sand, prime and paint those rough framing members. With some sanding and filling and proper priming, they may look OK for paint. You can cap off the top with anything that looks good to you.
#3)
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Apply the inside stringer covers first (whether is be sheeting stained to match or actual flooring)..
I would'nt try to use flooring to install on your stringers. That will look just plain weird and be hard to pull off..When installing the flooring on the steps, carefully measure how much the flooring will need to be ripped. It always needs to be ripped down. And install the first ripped piece up against the riser, tongue facing out. If you're gluing, no need to face nail unless you want to. Also, if you're gluing, just nail enough to secure the boards untill the adhesive sets and cures (24 hours) Now if you measured accurately, the last board will be a full board and the groove of the nosing piece will engage the tongue. I apply a little carpenter's glue in that groove and glue the nosing down to the tread as well. Then a few well placed headless brads just to secure it untill the glue dries well. This way, very little puttying is needed and the adhesive is stronger than nails anyway, most of the time.