Hi The right way to do it and avoid other problems, is to sand the floor, stain it in the color you like (I will go with light color; when the floor will be scratch you not going to really see it unless its really deep in to the wood) and use good finish that will protect the floor (I will recommend on the Dura Seal X-Terra http://www.duraseal.com/sections/products/fp/x_terra-fin.html its strong and will protect the floor from the cats and the kids Here is a link that might be useful: Dura Seal X-Terra

I'm curious to see the replies. My son almost bought an older house, 1951, the wood under the carpet is about 2"wide strips looks like solid wood, but seems to have a lot of nails. Is that how it was done back then? I wondered how it would look if he had them finished, but he backed out of the deal after the inspection. We are about to tear out decent wood because the sun has faded it and since it is not a big area it would cost a lot to refinish, and I hear the refinished floor would need to be done again much sooner than a new prefinished. I wish I could take out the old wood in a way that it could be used by someone, but I think it will get wrecked. The store I am using charges $3 sf to install it, the wood price obviously varies. They use their own employees, no subs, and did a nice job on our mbr and hall, so I will use them even though I dont know what an independent installer would charge. My maple will cost $5.54 sf, plus install.

"Tom Silva replies: You can do it either way. My painting contractor, Jim Clark, wants to be last so doesn't have to repaint baseboards that have been splotched with stain or ployurethane, or nicked by a sander. Besides, even the most careful flooring contractors inevitably leave a film of dust everywhere, he says. My flooring contractor, Pat Hunt, sees things a little differently. He prefers to be last because floor finishes take a while to fully cure, so he doesn't want anyone walking on them even when they're covered with protective tarps. He also says it's pretty common for other contractors to damage or spill stuff on a finished floor. The way he figures it, it's much easier to touch up paint than a floor finish. As the general contractor, I sometimes have the flooring finisher go first, and sometimes I let him go last. It all depends on what's best for the project schedule. I have that kind of flexibility because I know Clark and Hunt will put a lot of effort into protecting the other guy's work." https://www.thisoldhouse.com/ask-toh/what-comes-first-wall-painting-or-floor-refinishing

When I hear someone recount something that makes no sense, I generally figure that someone misspoke or didn't hear correctly. Whenever I see huge variations (3x) in price quotes, I generally figure someone made a mistake. Maybe the third quote will clear things up. Whether wood can be refinished depends on the type of floor and condition. Since you're selling, and already having work done in the house, I'd be inclined to have all the floors redone unless the finisher can match it 90%. If there's any doubt, do it all. This assumes you know what prospective buyers are going to want. I usually recommend the opposite when people are living in the house. Colors and preferences here are all over the place. Usually it seems foolish to upend residents and spend their money when there's a good likelihood the next person won't like their choice, but your situation sounds different since you're already having work done. The only warnings I have about sanding painted floors are that you may find some cosmetic damage and you may end up with paint residual in some cracks.