Just curious, what are using this for. Your original pdf is Vector while a Transparent Background png is now Rasterized.
That fine of type or any type for that matter should be Vector when possible.
Your type is also a 4/color build. If this to be printed it should be 100 Black only.
Yes it is.
What program are you using to add this Logo to the watches?
Use the file you have and set it to Overprint or Multiply. The Visually seen White will disappear.
If the manufacturer is doing it ask the what file format they require.