Unkown if the assignments were actually in the backup, the restore might be reporting the truth. It restored what it could restore.
One way to know for sure. Take the backup file used to restore and change the extension to .zip. Unzip it. In the extraction, look for an .xml file related to assignments. Open that file with a text editor (ie, NotePad) and inspect. If there is nothing related to assignments in the extracted backup of the course, there was nothing to restore.
Check: moodledata/temp/backup/ for .log files. .log files of larger than 0 byte size might contain clues. *Might* be notifications of something like 'could not find handler'.
1. ??? What option to do what? are you referring to the 'Assignment Upgrade Helper'?
2. Guess: the server version 2.5 you took this backup from has not converted Assignments from the old 2.2 to the new mod? Could your local install not have both Assignment and Assignment 2.2 enabled?
3. Answer: ? not really enough information to go on.
'spirit of sharing', Ken