I have a layout with transparent parts. How can I make sure that this parts will be stil transparent after printing? Which format should I use (currently it is .ai)?
This will depend on the art: if it is all type and solid colors, probably AI/EPS/vector. Check with the print supplier.
Note that "normally," white is actually the paper color. For a "clear paper," you will need an actual white swatch.
Almost any file destined for print these days should be saved as a PDF/X-1a:2001 file regardless of the actual printing substrate.
However, ask your print provider to be certain.