Before I can unmount an SD card, do I need to call any function to stop it?
You need to close any open files.
If I insert the card again, do I need to start it before reading from it?
No. The device has no way of knowing that the card has been inserted/removed.
If I have opened a file to write on the SD card and then remove the SD card, when I insert the card again, will I be able to just continue to write on the same file or do I need to reopen it?
Neither. Removing the card without closing the file corrupted it.
I wish someone has answered my question about the whereabout of the SD library manual.
There are a number of variations of "the SD library". You weren't specific enough.
In any case, nothing that the library does differs in any radical way from reading/writing files on a PC, except that the media is removable.