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...
If your floor was finished in the last few decades, it is likely that the finish is polyurethane. Do not wax polyurethane! There are a number of cleaners made for polyurethane floors that may remove some surface material that dulls a finish, but does not leave a residue.
Large area - If the finish in the large areas is dulled from abrasion, the only treatment that will restore the overall shine level is screening the whole room (or at least a large defined area up to an edge where a slight shift in gloss will not offend).
This is a process of slight abrasion using a commercial buffing machine (rented from a home center) that takes off the gloss of the whole floor. It is much less intense than a complete sanding and refinishing. The floor is then cleaned thoroughly and refinished with a new topcoat of polyurethane.
Paint globs - The paint globs must be chipped up. You could try a flexible bladed putty knife slid under the edge. If that is not successful, scraping carefully and gently on the top of the paint blob until you just get down to the finish, sometimes the thin last layer will flake away from the finish without much damage.
If the finish is damaged under the blobs, a light sanding followed by a feathered brushing of new polyurethane (of the same gloss as the floor) should cure it. It might be a bit shinier that the surrounding area, but it will dull a bit over time, or you could dull it down after it is well cured with a gently rub of an abrasive dish pad.
Stair edges - The stair edges should be sanded overall, onto the stair tread by about 1". You can use masking tape to make the sanding and subsequent refinishing straight and even. Use several coats of new finish on the edge. When dry, remove the tape. The transition from the old finish to the new on a stair edge will not be very noticeable. Again, you can use a mildly abrasive pad to soften the transition.
Best Answer
You might try a glaze coat and then over-seal with more varnish.
Some background:
A pigmented stain is made from 'bits' of color (the pigment). When applied to bare wood, most of the color is on the surface. (But HerrBag, I hear you saying, my Minwax stain is oil based and penetrates. Alas, Minwax is a MIX of dye and pigmented stain.
Most paint is pigment mixed into a base material that lets you disperse it evenly. The opaqueness comes from the concentration of pigment and the base material itself.
GLAZES
are the same pigment in a transparent base that lets you 'float' a color layer over previously finished wood. Indeed, to make a custom glaze, you add a small percentage of regular paint to the glazing base.
To adapt the technique for large areas will require the slower setting version glaze and a swivel head applicator (as used by flooring pros, its an oversized tubular paint pad (like a paint roller that doesn't rotate)).
A good article on glazes and glazing from Woodworkers Journal is here.