Like Jerome, I wouldn't try to clean the medals. Mount them as-is.
My favorite method of medal mounting is to use an overlay of clear film (AKA Mylar-D or Melinex 516). There should be detailed instructions in the archives, but here's the short version:
Trace a template of the medal's shape on a piece of scrap matboard, and draw three or four 1/8" wide "legs" extending away from the perimeter of the shape. These will become straps which are pased through the mount board, to hold the medal securely.
Next, tape a piece of clear film over the template, which is at least 1-1/2" larger all around. Trim out the shape with a #11 X-Acto knife (or your choice), including the "legs" you drew.
Lay the clear film overlay in place on the background board, and mark where each "leg" should pass through. Cut the slots just inside the perimeter, so they won't be visible after the mount is done.
Fit the medal in place under the overlay, pass the straps through, and secure them with 3M #889 double sided tape.
The pinbacks can be secured by trimming a slip of clear film to the inside width of the pinback. Loop it over the closed pin and pass it through a slot in the mount board; secure it with 3M #889 double-sided tape. (ATG & most other adhesives will not work on clear film, by the way.)
This method gives very good support to the heavy part of the medal, removing stress on the pinback and fabric. Also, the medal can't move around and mar the background board -- a real plus if you're using suede background.