Belt sanders are rough tools and the end result is never going to be good enough for polyurethane.
Use your sander carefully to get the tabletop as flat as you can, then switch to either a random orbit sander or sanding by hand.
Hand sanding gives better results, but it's slow. Wrap the paper over a block of wood especially for rougher sanding.
Now that I've finished sanding, there are quite a few spots that have white splotches from the wood filler
That looks like something you get almost always when sanding wood that isn't absolutely perfectly flat — fine sanding dust settled into some minor surface imperfections (they can be incredibly small or shallow and still show this effect).
To check if ...
For fine woodworking, I understand you should let wood dry and get accustomed to your shop (or ideally wherever the finished product will live). Okay so of course that make sense for a huge slab of wood, but is that necessary (or even smart?) for gluing up cheap pine 2x4s?
Yes, not just smart but many consider it mandatory. This is partly based on the wood, ...
How do I finish my piece now? Do I sand again and re-apply water-based Poly?
What are the silver/gray marks and how do I get rid of them?
As the silvery areas only appeared after using the steel wool, and seem to disappear when wetted, a fresh coat of finish may be all that's required to remove them1. But see last point.
Used 000 steel wool to get rid ...