First and foremost, welcome to *A!
Next, you should go to the *A website and download the *A timetable. Much easy than the 240 pages you printed out.
Now, as for the rules.... They varry a bit by carrier. Rule of thumb is that "most direct routing" applies, though you may be able to work around this.
I was able to get a ticket as one award that had this routing:
LAX-NRT-BKK (NH, F)
Destination
BKK-PVG (TG, C)
Stopover
PVG-NRT (NH, C)
Under 24 hour overnight connection
NRT-SFO-LAX (UA, F)
I got another ticket like this:
LAX-FRA (LH, F)
Under 24 hour overnight connection
FRA-TLV (LH,C)
Destination
TLV-FRA-CDG (LH, C)
Stopover
CDG-SFO-LAX (UA,F)