You have to stain with all the inlays and mounts in place. Otherwise, the inlets will swell and you will have to re-cut the outline of the inlet in order to get the inlay or mount back in place. Additionally, you must make sure that the wood is absolutely level with the inlay before staining. If the inlay is down in the wood a bit, the wood will swell over it and you will get a chip out if you attempt to remove the inlay. This is an issue for me because I do a lot of iron mounted guns and must remove all the mounts and inlays after staining in order to polish and finish the metal. I put the final finish, usually oil, over both the finished wood and metal.
