removing the door to kitchen would be easiest & cheapest way to open up & brighten the hall area. Next best, but more costly would be replacing door with one with glass panels. Look for used one in 2nd hand or @ your local Habitat for Humanity store. Keep the one you take off just in case other one gets damaged and needs to be replaced. Replacing only two panels would not achieve much I don't think. A mirror should only be placed where it can reflect something of interest & expand the room. If I recall, the staircase is what would be reflected? A photo collage frame? I think more colour is called for here. Your home, your choice. Bottom line is you decide what is right for your space. As you have likely noticed everyone has a different opinion, professional or not. Most important in re-styling a space is what the home owner wants to use the space for & to be their taste.

Carpet on the stairs will help with sound! But if stairs are good enough, just paint them all white! Add a stencil on the riser - numbers or a saying is always fun...or even add family member names. Do you like bold colors? If so, the handrail wall could be a green apple. Something that works w the pretty stain glass. Also upstairs, there looks to be space to make that a built in bookcase. As for under the stairs, you can take out studs but you can build deep shelves and give it a stair step look. If you have large gym bags, make some large enough to fit sliding them in. Whenever we put in storage like that, we add plugs for iPads and cellphones. Easy way to remember to charge them! Also, a narrow bench on wall opposite storage will help w putting on shoes.

A niche would be a good way to add some drama without infringing on your circulation space. It doesn't have to be deep to be effective. You can do one simple vertical rectangle, and line the back with a mirror, or add some colour with paint, wallcovering or wood. It would be great if you could also add a recessed light fixture at the top. The attached images are perhaps too busy for your situation, but it gives you a good idea of what can be done. Good luck!

What a wonderful hall looks fantastic. For the carpet I would (budget allowing) use chrome or brass stair rods on each stair would look brilliant in there. You could always get an old or cheap console table & paint it yourself in silver or bronze etc.