I had the same problem. Had old strip flooring that was over 100 yrs old. At one point someone replaced a section with 3/4 thick wood floor but you wouldnt know until it was tore up. When I laid all the flooring an got to the the hallway I was up about 1/2 in. I shimmed up an 1/8 in then stripped a blank of the new floor on the table saw to build a thin transition piece. There is a slight raise but everything is the same level and looks nice. So in short just use a reducer.

your wood actually looks like doug fir. I don't know if they were meant to be covered. you can tell by the way they were installed and finished if they were meant to be stained. I'm guessing since they are raw, they weren't. It's paint grade wood. if you stain them, it will prob look like this you can always paint the risers and get oak treads and stain those. Or just give all the wood a very good sanding. clean up all the rough spots and sand smooth. use a vac to suck up every spec of dust and wipe w/a damp rag. now prime them. do a fine sand when dry because the primer will give the wood 'nubs'. you want those nice and smooth before painting. clean up very well and get every speck before you start painting. use this primer paint. Advance is beautiful on wood. you can also use Zinsser oil based primer. If you want to try staining, use a dark walnut (general finishes) Gel stain. then top w/general finishes

With the stairs being the geometry that they are, I would ask that the whole exposed end be wood (or painted MDF/HDF) If they do drywall there with a baseboard there is going to be just a little sliver of wall paint left and the baseboard dying into the last carpeted riser as it meets the floor is not going to be a good look. The skirt on the stairs as it gets to the last outside corner can then turn vertical and attach to the baseboard adjacent to the door. Have you been walking around the house quite a bit already? Without a railing or newel post to mark those last 2 steps I would be concerned with coming out of the room next to the stairs and tripping over the bottom step.

The post I mentioned would not be at the bottom tread so the design of a continuous handrail would have to bypass the post to get to a newel at the bottom tread and there would need to be a guard at the new opening whether it be solid, balusters, etc. At first glance it seems a potentially messy solution. Perhaps the handrail could be at the other side and the guard could terminate at the post. The bottom tread should have had returns at each side. Its looks like a abasement stair in the entryway.