1) the FP can be sent anywhere in the process, but usually before/after the I140. There is NO prescribed method to the madness.. I got my FP notice, 4 months before my I140 notice, some other collgeaues (who got the GC around the same time) had received the FP notice almost 1 year before my FP notice.
2) You will not be informed about the name check. Your name is checked against the FBI list of wanted criminals. It usually happens once the FBI has your finger prints.