you'd be better off using a colour that doesn't exist within the standard palette for index 0 (bright green, or pink for example) rather than black. GIMP may be giving you false results with the eyedropper tool. i use "transparent" myself.
as for the way the colours are treated within the sprite, yes, only the indexes matter.
yes, i used a batch operation:
open file,
convert to RGB, (optional step, this step was unnecessary for yours)
convert to 256 colour and import palette from file, (i have these saved separately as .act files, readable by photoshop)
select colour range from pixel 0x0,
delete selection, (my .act files contain "transparent" as colour index 0)
save file,
close file.
granted this isn't a foolproof method, but it takes 99% of the work out of it for me.