There is really only one way to rid yourselves of the "white" fill spots. You will need to carefully dig out a small amount of the filler so only dark wood is showing around the edges. You don't need to remove all the filler from the original hole. Buy a pre-colored filler crayon, Minwax and others make them. Match the color of the filler stick to your floor color. Sand a small area around your repair with fine sandpaper, an inch or two is plenty. Fill the indent with the matching colored filler, smooth it well and put a couple of coats of urethane over the repair, feathering the urethane into the existing urethane finish. A foam brush would be good for this purpose. Good luck.
If you're staring with raw planks... PLEASE apply the same finish to all surfaces (underside as well as topside). Otherwise, the surface with the lesser finish will tend to absorb/lose moisture much more quickly than the surface with the heavier finish, and the planks will "cup", "bow", and warp with seasonal humidity changes.
Better than starting with raw planks would be starting with tongue-and-groove planks. The tongue/groove joints helps lock each pair of adjacent planks together so they commonly bear the weight of people & furniture. The T&G joint also helps prevent dirt from dropping down between dried/shrunken planks, preventing the joints from closing again in humid weather & causing buckling of the floor. Last, it helps stop breezes from blowing up through the floor from the space below.
It's possible to install the new flooring as a "floating" (not nailed down) floor over foam insulation (mostly for thermal insulation, but there IS a little sound insulation value)... but that's a whole different type of floor and really begs for a completely different approach. If you try to install any resilient insulation under nailed planks, you'll just get "nail pop", where the nails pop up through the surface of the floor and trip you, possibly tearing bare feet. There are completely different materials available (google "pergo") for floating-floor applications.
If it were my floor... I'd leave all the original floor planks in place regardless of condition, and cover them with all new flooring, diagonal or perpendicular to the original planks' direction.
EDIT: ALWAYS allow new flooring to "season" in its destination room for several weeks prior to installation so it's in humidity equilibrium with the room, preferentially always install during the period of highest humidity and make all joints TIGHT, and ALWAYS leave at least 3/4" (18mm) gap around all edges. You'll probably need to remove any existing mopboards/baseboards & trim before flooring, then cover the gap with them after flooring.
Best Answer
You can certainly varnish a section or a spot. Far better than leaving it bare, if it is, where it will just get harder to fix while you wait - even if it does not match at all.
Making it blend is more trouble, though some of that can be helped by "just using it" rather than worrying about about it when it's freshly redone and the rest of the floor isn't - it will be far less noticeable after a month's wear, in most cases.
Since you mention "colors" I'm a bit concerned that you are dealing with not just varnish (which may vary from water-clear to various amber-yellow colors depending on formulation) but stain, which is a lot harder to match.