Valve’s Proton is open source but is it also free to use and distribute in commercial software?
Yes.
Valve’s Proton code is licensed under the BSD licence, which is a “do anything you like with this code” licence.
Wine code is under the LGPL. You can ship this in commercial software as long as you “make the source code available” (which, assuming the distributor isn’t modifying the Wine code further, can be achieved by just linking people back to the main Wine project code repository).
DXVK is licensed under zlib, which is functionally the same as the BSD licence.