You certainly want to sand back to the wood, leaving a flat surface, so that you've still got timbermate filling the pores but not so much that you sand off the layer of wood with the filled pores and expose more pores beneath it. It's a fine balance and often a single grain fill and sand isn't enough and you need to fill and sand two or three times.

However, the basswood itself doesn't need grain filling, so you're sanding down to the wood leaving the small dents and scratches filled. The body will be a basswood ply, with a thin veneer of figured basswood on the top. SO you need to be careful when sanding back not to go too far.

The neck is solid maple, so you're free to sand that as much as you want until you only have the Timbermate filling any dents there may be.

I'm not sure about the white marks. They seem to correspond to places where it looked like there were some glue marks in earlier pictures, so it may be how the Timbermate looks like on top of those patches. Worst case is that you've sanded too far in those areas and hit the glue layer between veneer plys. I've done that myself.

It's very hard to tell without having it in front of you, even with a video.