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...
You'll need to start over, sanding out the splotches of glue. To deal with the divots of glue, they can be 'set' using a nail-set in the same manner that you'd set a nail ... or the divots of glue can be dug/scraped out ... or simply left as-is.
You'll likely have similar splotches with store bought wood filler, and blotches can arise from other contaminants as well. One option is to re-sand, removing the existing splotches of glue, then apply a coat or two of the poly. This will seal the wood, then set/refill the divots if/as desired. As an alternative treatment, the divots can be blended in with a faux-finish by using a small artist's brush and a few tubes of artist's paints. Then apply your final coat(s).
Sorry for the bad news. Glues and fillers are a bane to fine woodworking. Take a week or two for a breather and regain your motivation to do a good job.
Best Answer
It is no problem and probably a good idea to use thinner to wipe up the stained poly. I would think the tricky part is going to be the tendency for the grain to get raised (rough) on the bare stained part. I think you will have to buff again but I might put 2 coats of poly down first. Then buff with a fine screen >=120 and put a final coat.
Be sure to turn off the pilot lights and watch out for anything that could flash that vapor.
I'd give the stain overnight to dry or at least til it is dry to the touch otherwise the stain will leak into the poly and muddy up the finish.
I've switched to water-based poly. Its expensive but it dries real fast and it now outperforms oil based. No companies are putting research money into VOC stuff and they suck for the environment.