I am not sure I am completely understanding all your questions but I will try to answer based on my interpretation of your questions:

1) You can book a stopover on a JAL award through Japan using Alaska miles, assuming you will be allowed to enter Japan. You cannot backtrack though. No SFO-TYO-KIX, stop, KIX-TYO-BKK. You can do SFO-TYO, stop, TYO-SIN/BKK. Any length of stopover that is bookable in the open Alaska calendar window is allowed. The calendar runs for 330 days so if you book today both flight segments would have to be completed within 330 days from today. You could book a flight to TYO departing today with around a 328 ish day stopover before flying from TYO-SIN/BKK. So, virtually unlimited stopover duration.

2) No you do not need ITA matrix to book a JAL award using AS miles and as a matter of fact ITA matrix is useless for this since it does not search award inventory. Use AS to search for the flights (or other partners, AA is especially good since you can use the calendar award search to find one-way non-stop awards). You probably do not need to even call AS to book. Once you find availability for the flights you should be able to click on the "All search options", select multi-city search, use miles, put in your route and dates for both segments, and the search should allow you to book it online. Not sure if online works if you arrive and depart from a different airport in Tokyo (like arrive HND, depart NRT), but phone agents, I think should be able to handle that.

3) Expert Flyer is not much help. It will not search for JAL award inventory so you probably do not need Expert Flyer for this.

4) Yes, although there is a website bug that does not show J availability if returning a trip with an F flight in it so you will have to use workarounds to book one-way flights that have both F and J segments. You can find details in previous posts in this forum.