The white residue is abraded varnish, settled in the low spots. Ideally this should be rinsed or wiped away as the final stage in the wet-sanding before moving on to the next step. A damp microfibre cloth is an excellent tool for doing this, but just flushing the surface with water and any cloth will get rid of it, just as everyone had to do before microfibres came along :-) I've even seen some finishers using their hand as a type of squeegee when wiping the surface down, although I'm not convinced this is better than using a cloth.
I have tried using mineral spirits to get it out. It looks fine with the mineral spirits on, but once it dries the white shows back up in the pores. This gives me hope that maybe if not completely cleaned out, it will not be so visible with a layer of finish.
Yes exactly so. When you apply varnish this is said to wet the surface and the effect as you've now seen is permanent. This is much the same as the difference between bare sanded wood, which looks pale and dry, and the richer colouring once the first coat of any finish goes on, wetting the surface fibres. Note that the wood is still said to be 'wet' by the finish after it has dried.
Some addition points:
Do all your sanding in one go
You can do all of your sanding at the end, so rather than sanding to prepare the surface for the final coats as you've referred to you are smoothing off and perfecting ("flatting off") the surface of the varnish.
Sanding between coats isn't necessary unless you're removing dust nibs or other defects — on most finishes it is not needed to ensure bonding between one coat and the next. All excessive sanding between coats is doing is removing finish you don't need to remove and slowing down the whole process.
Bit more on this here:
Leveling a finish/finishing the finish
Grain fill/pore fill
You can fill the wood prior to applying the final finish to minimise the potential for this problem. If the wood's surface is very flat and smooth to begin with you won't easily have any low spots that can harbour sanding residue.
With varnish, which can have a very substantial body so it builds thickness well, you can fill grain using the finish itself (often done by luthiers for example, usually with shellac in that case but the same principle applies with most film finishes).
Another option is to sort of do both things at once, by sanding in the first coat of varnish. This creates a slurry of wood dust and varnish, which is an excellent grain filler and of course you have no worries about any compatibility issues between the fill material and your final finish since they share the same binder.