"Will we be able to get the stain out of the wood?
Any idea how deep it might have saturated?"

Depends on what type if "stain" it is.
The common stuff is pigment stain, and really just thin paint.
Penetration is very shallow (1/32 inch).

"Would we first remove the polyurethane with some kind of chemical that would "melt" it off, and then SAND the stain out?"

The chemicals to remove polyurethane are usually pretty harsh.

You want to start with a chemical stripper. Methylene Chloride is the most common, fastest working, strongest, and unfortunately, most noxious. Put it on and let it work. Do a few steps at a time. If you have nice weather, open the windows for a breeze. The biggest problem people have is not giving it enough time and then having to spend more time and effort on a second, third, fourth ... application. Scrape off with a putty knife and follow up with a scotch-brite stripping pad. Neutralize / rinse as directed on the can of stripper. Then let the wood dry thoroughly. This will remove the polyurethane and generally most of the stain. Oak, having coarse pores in the spring-wood, might retain some of the stain in these rings. Don't kill yourself trying to get it out, just leave it as a contrast.

Often people think they can just "sand off" the finish and avoid the chemicals. Bad idea. After a whole weekend and a pile of gummed up sandpaper, and only part way through the project, they think it's a bad idea, too. The other risks are that the finish is removed unevenly (causing the subsequent stain to absorb unevenly) and if there is veneer (not likely on your steps, but highly likely on casegoods), it's easy to sand through. An exception to this guideline is if you are doing a hardwood floor with a drum sander and removing significant amounts of wood with 36 grit belts.

Once dry, do a light sanding with a fine sandpaper (120-220).

I think you might want a stain, something like a "golden oak" if for no other reason than to even out the tones in the various pieces of wood. If you want to preview what the unstained wood with varnish only would look like, simply wipe some areas with a rag dampened with mineral spirits.

I would also recommend applying a good floor-quality polyurethane. This is one instance where you need the abrasion resistance of a poly. Get a good brand, not just a can of polyurethane at the grocery store. Just because it says "Great for Floors" on the front of the cans mean that it's the best choice. If you want, you can thin it 50-50 with mineral spirits, creating your own "wiping varnish" and wipe it on. You will need 2-3 times as many coats as a brushed on finish, but it's fairly foolproof as a DIY or beginner. I also like to sand between coats to smooth it all out, remove imperfections and dust nibs. If you are wiping on, every other coat should be OK. If you are brushing, be sure to thin it so it flows out better.

Thank you brickeyee and bobsmyuncle.

Brickeyee, I found out it is Duraseal oil base stain. Coffee color (dark).

Bobsmyuncle (love your name), I appreciate the step by step, and am quite sure we are not up to this, but it is good to know so that whoever we hire, we'll know what he's supposed to be doing.

There is carpet on the stairs, wrought iron balusters, painted step trim, and other surrounding things we don't want harmed. Will it be safe if carefully covered with plastic and taped, or will the stripper etc. eat through that if it drips down on it?

Any stripper I've used will not eat through plastic or the right masking tape. Apply the tape carefully, burnish it down and be careful with the stripper.

Here is a link that might be useful: 3m 2060 lacquer masking tape

Just as an aside, I went to a finishing seminar yesterday. I was talking to the instructor prior to the seminar. He told the store owner that he'd run out of the varnish he was using on his floors and was backordered to get enough to do the last coat so he switched to a common consumer brand. He was really unhappy with the result and the application.

Bobs... Thanks re masking tape, who would have known??? About the plastic, any particular type or thickness? Everything else in the house will be covered and taped with painter's plastic and blue masking tape (for dust, as the floors are being done too).

Also, re the finish coat, I see what you're saying and would not use cheapo product of any kind. Save $100 and ruin a $300K house! NOT! (OK, it's probably a $250K house at this point.) Anyway, since this is just a handrailing and newel posts, none of which will get any heavy use, do you really see a problem using the Bartley's gel varnish? I've used it on a piece of furniture and #1 it's very easy and foolproof to put on, and #2 it really gives a beautiful beautiful glow to the furniture without any shininess.

I really appreciate your know-how and advice.

Thank you Brickeyee, that is crucial information.

We are now tending toward leaving the railing dark, replacing the wrought iron with traditional painted wood balusters (which I prefer anyway despite the popularity of the wrought iron) and painting the newel posts and balusters white. No stripping. Too messy, time consuming, and too much for us... plus what you just said about the noxious fumes, -- though no problems along those lines it just seems too much on top of it all.

We will have to fill in the oak grain with putty and sand the newel posts (or have a painter do so), and any info on that will be appreciated. To me it sounds like: sand it lightly, apply the putty/woodfill (I'm thinking a little soupy on the turned parts), let dry, and sand.

Does that sound about right?

A painter we had over said he could paint the newel posts enough times so that the several coats would even out the oak grain. Any idea if this is possible?

Balusters will come primed, so that seems easy enough. I thought I'd buy them online. Builder told us (and we've taken one out) that balusters are installed AFTER handrail and can be taken out/put back in without removing handrail, and sure enough if it isn't true.

I have similar situation my stairs are carpeted my railing is rot iron and then it’s all dark stained we plan on ripping up the carpet on the stairs and upstairs in the bedrooms. I want to go with a light grey floor but my husband is telling its impossible because of the dark stain we have and that we have to go with a dark wood upstairs to match the stairs. I done want to do floors I’m not happy with and I done want to just settle with dark wood. He’s saying it would cost thousands of dollars to sand and redo all around the dark stain to get it light ? He says it’s a big job and will cost way to much. Can someone explain to me if it can be done. If anyone knows a great company or handyman who would like to come and give me a quote and do this job would greatly appreciate! Thanks