First, a quick note on technique - start with a coarse grade sandpaper on the floor sander and work down to the the finer grade.
The coarse grade removes the old finish and levels out the imperfections, then the fine grade makes it nice and smooth. If you go the other way around, you'll definitely curse every time you see the grooves in the sunlight. Fixing this means stripping off the nice new finish, not fun.
In passing, an obvious-in-hindsight but sometimes-missed point is to go with the grain, i.e. sand along the boards, not across them. In any case your sander will complain bitterly if you don't.
Also, give the boards an incredibly good wash before you start with e.g. Sugar Soap - you want to try and strip off everything gungy to stop the sandpaper clogging too often. Leave for at least 24 hours to dry nicely.
Be sure to hire an "edge sander" too, this will help you get to the edges that a drum sander won't get to. If you can, remove the skirting boards (baseboards) as then you can get right underneath; only do this if you know you're redecorating the walls too and don't mind patching up the woodwork. You'll inevitably end up doing the fiddly bits around the door frame and right in the corners yourself by hand with a scraper or sandpaper, which takes longer than you could possibly imagine.
While doing the sanding, remove everything from the room (obviously) and get heavyweight dust sheets to tape around the door. Wear airtight goggles as this is an insanely dusty process. You'll also need ear defenders as this is an insanely loud process too.
Regarding stain, it's a matter of personal preference. I've done both and prefer the more uniform result that stain gives.
Regarding finish, I've used both oil-based and water-based finishes. Both are OK, but the oil-based one looked richer (can't say why, can't say how) -- and took about three days to dry. Somewhat inconvenient for a hall, and our puppy left nice pawprints.
Follow the instructions on the tin; in general thinning probably isn't worth it, as you'll find the second and subsequent coats will go further anyway. I applied three coats of the oil-based finish, and five of the water-based finish, but this depends on the state of your boards. It's handy to rent good bright work lights so you can see where you've been, especially for the second and subsequent coats.
What is important is that you get flooring-grade varnish, regular varnish won't stand up to the first party, and yacht varnish will cost so much you won't have the first party...
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Before doing anything else, give part of the new board a swipe with paint thinner. Dampening it gives you an idea of what it will look like under a clear varnish, and paint thinner evaporates quickly enough that it doesn't tend to raise grain or otherwise interfere with the rest of the finishing process. That'll give you some idea of how much the color really needs to shift.
If you have some scraps of the old floor and of the new wood, a good paint store may be willing to dab the latter with samples of several different stains to see what they'd recommend as a good match. If in doubt, remember that you can always apply more stain in successive passes.
In my case -- and I'm not sure this is a recommendation, just an observation -- I went with a very cautious approach. Oil varnishes, especially older ones, tend(ed) to add a strong yellow tone to the wood. I was going to use water-based poly, which hardens clear. I applied multiple passes of dewaxed shellac (which I happened to have on hand) to the oak thresholds to achieve that color shift, then applied water-based poly as the protective layers. I stopped short of getting a very close match, but it's close enough that the thresholds don't stand out. I'm sure a real stain would have gotten me there faster and more exactly, but this was good enough for my purposes.