The label will be printed on black fabric. How to change all black color elements into the transparent background, leaving only white elements.

Convert black strokes with Path->Stroke to Path - then Path->Union - then Path->Break Apart - select all black elements and hit backspace to delete.

not works for me. Do I need first Edit -> Select All,  Objects -> Ungroup? Or just start straight from Path->Stroke to Path?

I can't look into your file right now. If it's a group of strokes ungroup first - if it's a group of Color filled shapes ungroup first - if it's a compound Path already just start with Path->Break Apart.

I suggest exploring the Tutorials in Inkscape's Help menu.

Basic function are covered there and in the manual.

As suggested by polygon : 

• first ungroup all groups and unlink all clones
• all your lines are true lines : convert them to shapes (path > stroke to path). Then unite them all (select them all a do path : union).
• then you can break this path a part or do a difference with a rectangle :
  • draw a white filled rectangle wider than your logo, align both vertically and horizontally
  • send rectangle to bottom (windows or linux : hit End key)
  • select both and do path > difference

Edit : Sorry but I won't share the finished work as I just googled it and I don't share theses ideas.