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...
Staining will highlight any imperfections in the wood. From the photo it looks like something has been wiped across the grain. That could be when you wiped the sawdust away or possibly if you wiped the stain across the grain.
Having a good surface to stain is pretty much essential as the stain will highlight even the smallest imperfections.
Applying thin coats as John suggests is the best way.
Also, don't forget that the wood will age - especially for something left outside - so any imperfections you see now will blend over time.
Best Answer
1) As far as Mineral Spirits are concerned, you can apply stain as soon as the wood appears to be "dry" again (it will look "wet" until the mineral spirits have evaporated). However, see #3.
2) Most likely -- stain typically contains a thinner as a carrier, which dries more quickly in warmer temperatures. The age of the stain should not be a factor, unless it has lost the thinning agent due to the container not being sealed well -- this would make the stain appear darker. You can add back thinner to counter-act that. Thinner, in this case, is most likely mineral spirits -- you can use a retarder to slow the drying process if necessary.
3) The pre-stain more than likely has dried, curing the resins. A light sanding and a re-application within 15 minutes or so of staining should give you better results -- trying to re-stain after the drying and without sanding may cause the pre-stain to reject more of the stain than you'd like.