if its a looped-pile or mixed loop/cut, then its improper way to waterfall a stair system. looped carpets need to be edge turned. they will wear out and fray if they arent. Call the retailer/contractor back and get them to replace them. If they are cut pile, then that is standard, unless the skirt boards are out of square or unparallel to each other, then they may need to be edge turned to keep from excessive gapping where the carpet meets the skirt boards. IF you have gaps, that that is probably the cuplrit and they should have edge turned. That is also standard. A big mistable many neophyte carpet installers make with steps is not pre-measuring the stringer board widths to ensure they are square with the steps verticle plane and parallel to each other. If they give an excuse that any of the above is because of your steps condition(s), then they are uneducated hacks or they are lying to you.

Well, if you decide to stay with the solid tread/risers (IMHO... the best, but I'm biased), and you remove the side boards... the best application would be to replace the treads, because if you keep them, the exposed ends would need to have, what we call "returns", and it's probably not feasible to add them to the existing treads.

I don't see how that can be PVC primer. I'm no expert, just a DIY, but I've never seen bright blue PVC primer. My first thought was blue RTV sealant. But the way it's apparently run down one wall doesn't suggest RTV. It looks more like some kind of paint. The marks on the carpet also suggest that he set something that had been coated with whatever it is right on the carpet. What was he thinking? Regardless of what it is or what he was thinking, there's no excuse for this kind of sloppy, shoddy work. Serious question - do you think that the installer might have been drunk or high? You don't really say what you've done about this. You probably should have called the HVAC company and raised holy howdy the day after he left. If it's been a few weeks, they can probably argue that something else caused it. Hey, you have a toddler, so they can always blame the kid. :( Still, it can't hurt to have a quick chat with your lawyer. Sometimes a phone call from a suit is all it takes to get a company's attention. The common perception is that the low bidder will usually do a poor job. My experience has been that many factors affect the bid, and that there isn't anything close to 100% inverse correlation between the bid amount and the job quality. Notably, you can get solid gold references and still end up with a crummy job.

Will the flooring be the same on both levels? Do you know what wood is under the stair carpet? You might be pleasantly surprised to find oak :-) One other thought would be to paint both the treads and risers white to match your trim and put in a patterned runner (just be sure the carpet installer knows how to make the pattern match on the corner!) In pic below imagine the treads in the same white - the runner makes the stairs pop. Also, check your building code - your railing may need to be raised if it's original to the home. Good luck with your project!