Your current handrail is called "colonial" and your current balusters are known as 5015s. These are the least expensive stair parts back in the 80s, 90s and today. It is the cheapest a builder can put in a home. Your treads are painted on the exposed end and fully carpet on the upper flight so, your upper treads are most likely pine or even wafer board. Your lower treads if carpeted like that from when the house was built are probably "hybrid" treads meaning the exposed end MIGHT be oak only for 10" or 12" to under the carpet. The rest of the tread would be made of scrap wood (pine) so removing carpet to get fully stained treads is not an option. If you wanted solid wood stainable treads all the way up your stairs, the treads and most likely the risers will have to be replaced. It can be done but at a cost of course. Please upgrade your handrail to a different profile. Any good stair company will have 3 or more profiles that they use and if you look online, you will find 20 or more styles to choose from.

As for the baluster having to be replaced at 4" spacing per code, IF you just remove and replace the balusters without rebuilding the treads and risers, you most likely are grandfathered in to simply putting iron balusters back in the same holes and it would be acceptable BUT, I suggest bringing it up to code as it will be safer and be easier to sell if you ever want to.

I also noticed that where your balusters go into the wood on the bottom, you do not have traditional "landing tread" which has a rounded nosing overhanging to mimic the same look as the treads have hanging over the risers. That would be a nice feature to have done and it would be stainable just like the treads.

All of these small details tie into each other so that when done and done properly, you have totally transformed the look of your Foyer. It makes a huge difference and it will be something you will enjoy and smile about every time you walk by them or up them.