Matching stains is only done by trial and error---and having several scrap pieces is important. If you do not have scraps, you have nothing with which to try different stains or combinations.

So, first, you need to find several pieces of oak similar to the stair treads. There are two basic kinds of oak used in America---white and red. Some red oak can be almost white, but white oak is seldom as dark naturally as red oak. The age of the oak is also a factor---older oak may have a tighter grain pattern(as in the 1952 oak) than later wood(as in the wood you will find in a lumberyard).

If the new floor is oak stained Brazilian Cherry, the matching will be much easier. If the floor is a different wood, matching color will be very difficult.

The finish on the floor is another factor in matching, as some finishes change the stained wood color.

As for the scratchs, if you sand with 100 grit(using a random orbit sander---not a 1/2 or 1/4 sheet finish sander or a hand block) and then with 150 grit, they should disappear. Do not use finer than 220 grit, as thet can burnish the wood and it will not take stain well.

For finishing the stairs, I recommend MinWax floor polyurethane. The down side to using that is the cure time is a minimum of 72 hours before any traffic and a week for normal traffic. The oil based poly also adds an amber tint to the finish---as does any oil based finish such as spar varnish or regular varnish.

So, you need to find test pieces, sand them the same as the treads, and experiment with several recipes---starting with Brazilian Chery(or what is closest on oak on the store test examples---don't use the examples of pine or other woods as a guide).

If the first piece is too light, you can restain with a slightly darker stain. If it is too light, cut the stain with natural(clear) stain. Light sanding can lighten a color as well.

Thank you handymac. I did use an orbital sander on the scratches after a 1/4 sheet. I cannot imagine what was dragged up the stairs when the house was built; but it definitely left a unique scratch. Larger flaws sanded out.
Have you ever used the Minwax super fast drying polyurethane for floors?

The MinWax I used(oil base---for the added color enhancement) dried for reapplication in less than 12 hours and was walkable (bare/stocking feet) in 24 hours. I actually did 3 applications---one at 9PM, the next at 6AM and the third at 5PM. Then let it cure for 7 days(before putting furniture in the rooms).

Needs 30 days before adding area rugs.

I bought it at a Sherwin Williams store(SW owns MinWax.) That means they warrantee the product in case of problems.