No. If you have a process of the static binary already running, it will not be loaded again from disk but the same physical memory will simply be mapped to a new virtual address space for the new process. That's the time to set up a few MMU tables and you're ready.
Um...no...only the R/O sections. Programs have initialized R/W spaces that once written are gone. There'd be no reason to have those sections marked
as COW, unless you already had someone sharing the page (like a forked copy). But any unrelated process likely wouldn't use a COW copy, as they'd need the pristine data as it was supposed to be when the program loaded.