I think the banisters are fine, in fact quite nice, unless it's a Victorian or Edwardian house in which case I would put spindles back in.
New carpet on the stairs. Go for something mid colour to hide the dirt. We have a tight loop Berber with a slight stripe, mid grey/dark beige, works well. Sisal doesn't wear well.
For the hallway, tiles are very practical but cold underfoot so are you a shoes off or shoes on household? If shoes off then I'd go with wood, vinyl or laminate.
If shoes on then tiles. You can get tiles that look like wood planks or tiles that look like natural stone, I'd probably choose some like that. Unless it's a Victorian/Edwardian house - if so maybe geometric tiles.
Again for hallway floor go with mid colours, and something with a bit of fleck in it to hide dirt. That's why tiles that replicate stone or wood are good because they have some streaks in them anyway.
I agree about keeping the understairs cupboard - you need it!
Depending on if you have any budget for this, you could swap the big horizontal radiator for a tall vertical one (it could go behind the front door for example) - that would make the hallway seem more spacious and give you space for a hallway table. Looks like you have electric points there so you could have a lamp on the hall table which looks lovely and welcoming in the evening.
I'd also put a big landscape mirror along the wall above the radiator, or the opposite wall. Will make it seem much lighter and wider.