I've booked (and mostly cancelled) literally dozens of flights in the past few months using American Airlines miles. It is (mostly) smooth and highly predictable (and, I must say, American Airlines probably has the most consumer-friendly policies of any US airline, except perhaps Alaska Airlines, when it comes to flexibility for changes/cancellations on award tickets - so good on them).
Usually it takes a day or so to get ticked when using miles.
Yes and no.
If your flight is entirely on American Airlines "metal" (just their planes, nobody else's), then ticketing should be instantaneous.
If your flight includes any segments on American Airlines PARTNER airlines, then ticketing generally happens within 24 hours. Initially your booking will show as "on request." That's because different airlines' systems only interact with each other in limited ways; to do non-trivial things (like book a ticket, assign a specific seat, etc.) it requires some time (and often a human is required to perform some tasks). Until you see your trip status change to "ticketed" you do NOT officially have a rock-solid booking.
You need to call American Airlines and talk with them. Set aside some time, it may take hours.
FWIW, I have had generally good luck lately with accepting their "callback" option rather than waiting on hold (2 recent experiences: The system said I would receive a call back within 2 hours; I got one call-back as soon as I hung up the phone; the other time I got a call back about 90 minutes later). Be patient, be prepared to wait a while. Try calling very late at night or super-early in the morning.
IMPORTANT: Look at your booking on the aa.com website. Does it pull up your detailed itinerary? If not, worry more. If yes, look to see if any flight segments are on partner airlines or listed as codeshares. Look for the Confirmation Code/Reservation Code/PNR numbers -- there should be one for your entire itinerary that's used by American Airlines. Below the AA number, look for one (or more) additional confirmation codes/reservation codes/PNR numbers for any other airline(s) on your itinerary. Every airline uses their own number -- their different systems usually can't "talk to each other" and when dealing directly with that airline, every airline uses their own system.
Make a note of any numbers for your flights on other airlines (always do this!!!), then contact that other airline directly, and give them their own number for your flight.
It's possible that your itinerary is fine, it has been ticketed, and you are good to go -- but that happy status is just not being shown in American Airlines' system. I am in that same boat: I booked flights from Seattle to Bangkok using AA miles for a trip in April. Flights are on AA's partner Japan Air Lines ("JAL"). Later, JAL cancelled my flight and moved both my outbound and my return flights by 24 hours. Neither AA nor JAL ever advised me of that, but I could see something went wrong (I tend to my flights obsessively). JAL has us all good, same flights just 1 day later, but on AA's website the status still shows "on request". With the JAL confirmation codes, I can log in to JAL's website, pick seats, even choose my meal (flying business class), and all is confirmed on JAL's end. AA has never updated our flight status on the website. I called and spoke with two AA agents, both just shrugged and said things look OK to them. So you may have a similar situation, where the AA partner airline is all thumbs up, but AA itself has failed to reflect that status. In that case, check with the partner airline directly since AA is dropping the ball.
I'm seeing this on multiple flights I have booked with AA, so it's not a one-off issue.
If your itinerary does NOT include any partner segments, then I would assume AA has just dropped the ball. Set aside some time and call them.
Hope that helps. Good luck.