I should add all other trim (crown, base, casing) in the house is white

I don't think it looks bad because it matches the floor & stair treads. You are the one looking at it every day though, so do what makes you happy. If all of your other trim is white, then it would also look good painted white. It will require some work though. You will have to sand the varnish off so the paint will stick. You will also have to be very careful when you paint so you won't get white paint on everything else. If you can afford it, pay an expert painter to do it for you. It is a beautiful staircase.

I am working on a set of stairs myself, so I've learned some terminology. Stair treads are the part you step on, stair risers are the vertical part between the treads (yours is the dark stained part), spindles are skinny pieces under the railing (yours look metal), and the newel post is the big post at the bottom. I'm not sure what the piece that runs up the wall is called. Someone else will have to enlighten us.

We just finished our stairs & I love the white trim with the dark wood & black. I also have white trim through out rest of house. I'd go white if I were you!

I also love the look of white risers. The only draw back being black scuff marks on the white if you don't use a runner or make everyone take their shoes off.