What size should I get for women's shoes size 8-9?

Medium fits women's size 8-10.5.

Does this item come with extra tungsten spikes?

yes, a couple extra.. Honestly, you probably won't need them. I wear this brand of shoe spikes for both work and non work all winter. They do the job, they last, they stay on your boots, and I can drive with them on. tile and carpet are OK but don't walk on wood floors.

Are these actually spikes or studs? It seems impossible to find real sharp spikes!

I would say they are more of a spike than a stud.

I have several flights of marble stairs to walk down before I get to the front door of our building, are these okay to wear going down the stairs?

I wouldn't wear them inside. All of these products, I think, are slippy on most indoor surfaces. I change into my shoes with these on as I leave a building and change into other shoes as I enter one.