I want it to get rid of crappy items like Sulfur and Copper Wire, but it always leaves at least one stack. Is there a trick?

Why yes! Normally, it never trims away your very last stack of anything, because we can't tell if it was something you wanted to keep (eg red/green wire). But if you set a logistics request for the item with a minimum set to zero (which is normally meaningless), the trimmer will know you really are happy for a minimum of zero slots and will clear them out.

This is different from setting a max of zero, which will auto-trash completely. The trimmer will only clean you out if the stacking is clearly inefficient and you are under space pressure.

I want to move lots of Rail (or landfill, or belt etc) from this chest to that chest and it keeps emptying my inventory! Help!

Temporarily turn off the Logistics requests and auto-trash checkbox (on the Logistics UI). If you find you do this a lot and it's annoying, there's a setting to decrease the number of slots being kept free (default: 10). There's another setting ("Consider fullness a key factor...") that prioritizes retaining fuller stacks - I've not played with it much, but if you feel adventurous, try increasing it, 1.2, 1.5, 2.0.