If you don't prepare your documents differently, then yes, the spot colors will print under/ over your other colors. If you don't want that, you will have to remove the areas either by erasing them or excluding them later in Acrobat/ your RIP software, but then you will also have to deal with trapping and overprint issues to avoid gaps.
I now have deleted the underlying parts and it looks good in PS. But when I export as PDF via Indesign, it doesn't look good. Do I need to save as a special format? (DCS 2.0 maybe?) Or do I have to adjust presets?