Normally the underlay should not be fitted on the riser at all and on the tread it should be taken in from the edge if the carpet about 3-5cm. The fitters should have known this. I think if you are not happy with the job there's not reason why you should have to live with it, the fitters should come back and do it right. It's not normal to see the underlay from the side.

You are very lucky - beautiful hallway! I wouldn't bother with the wainscotting - there is enough going on and you can still do the contrast below the chair rail with different colour paint. I see you have quite a few samples on the walls....I would do the trim, balusters/spindles, and door in the same colour as the white/cream in the floor tile. The other colour or colours on the wall should highlight the beautiful interior doors, the floor and the stained glass (and paint the radiator(s) the same colour as the wall behind it). For inspiration, I haven't found anything very helpful on Houzz yet, but if you go to the Farrow and Ball website and then go to their "inspiration" site and select "hallway", they have quite a few that are similar to yours. I found one that used Wimbourne White and Manor House Gray with a black and white checkerboard floor that looked lovely, another one that used Mizzle, one with Brinjal under the chair rail for something dramatic (which, from the look of your samples, you might like)..... Anyway, if I was clever-er I would be able to link them to here, but I'm not, so peruse them and see if there is anything that suits. Enjoy!

I'd just paint the newel posts white. You could paint the top of the post and handrail in a light-mid neutral if wanted (as shown in the pic but with wood!) or just do the whole lot white. I might consider putting in spindles at the bottom as I assume your front door is a few feet away? I have the same issue with my staircase which has been altered, different period and it has spindles, but you step through the front door and after a few feet you encounter a "solid barrier" to the hall which is the bottom of the stairs!

I think the relative 'darknes's of a house determines the perceived temperature of the paint used. I've used Slaked Lime before in a really light house but it doesn't quite work in a darker house, Portland Stone has a little more warmth and colour ot it. Neutrals are way harder than saturated colours!