I recently bought a flight from SFO-IAD through LAX on AA's website (not separately - both flights are under the same reservation). The SFO-LAX flight arrives at 20:56, but the LAX-IAD leg departs at 21:40. This is a very tight window, and less than what AA recommends on their website.
Since I did buy both tickets from AA (and therefore wouldn't be checking in or checking luggage at LAX), am I guaranteed to board the second flight (barring exceptional circumstances)?
You are not guaranteed to board, no, and it is unlikely the flight will wait for you.
Your connection is guaranteed in the sense that American has sold it to you as a single ticket (which means they at least consider it reasonably achievable if all goes well) and will honor it. It is within the minimum connection time or they wouldn't have sold the itinerary. But if there are delays and you don't make it before the boarding door closes, your primary recourse is a ticket on the next available flight out at no additional charge.
As best I can tell, American generally operates two flights a day on LAX-IAD, and yours is the second one, so if you miss the connection, your likely option is to wait 12 hours later until the next morning, or potentially take a late night flight out to CLT or DFW with an early morning connection to IAD. The feasibility of this will depend on your travel plans and the urgency of your need to be in IAD the next day: would being late mean a small annoyance or missing a once in a lifetime event?
You should also note American's policy for accommodations in such situations:
When your flight is canceled or a delay will cause you to miss your connection, we will rebook you on our next flight with available seats. If the delay or cancellation was caused by events within our control or you were diverted to another city, and we don’t board to your final destination before 11:59 p.m. local time, we'll arrange an overnight stay at your connecting airport, if available.
If the delay or cancellation is caused by events beyond our control (like weather) you are responsible for your own overnight accommodations, meals and incidental expenses. American Airlines agents may be able to help you find a hotel.
Absent something obvious like a mechanical problem with the plane causing a delay, the delay will likely be caused by events beyond their control, and any expenses from an overnight stay will be your responsibility (these may be covered by travel insurance or the insurance provided by some credit cards).
The times listed on the website you linked are intended for those arriving at the airport as new passengers, not connecting passengers, and do not apply in this case.
No, you aren't guaranteed to board. The second flight will endeavour to leave on time, regardless of the status of your incoming flight.
If you do miss the second flight, though, American Airlines will automatically book you in the next available flight. This could be the next day.
Note also that even if you do make it to the second flight, there is a chance that any checked-in luggage won't.
As others have stated, you are not guaranteed to board the flight on your booking; you are not even guaranteed to board any flight at all - the airline may decide to deny you boarding (although this would be very unlikely in your case).
What you should really ask about is how you'd make it to Washington if you miss your connection. As both flights are on the same reservation, AA will simply put you on the next available flight (and, yes, it may be the next day, moreover it may be to DCA, as once happened to me).
Generally, about 45 minutes for domestic to domestic connection on the same airline should be OK. The airline would not have sold you the two flights on the same ticket knowing that the connection is not possible. Anecdotally, I have managed just fine with an international to international connection in 45 minutes (in Munich) and have missed a 3-hour domestic-to-domestic connection (in Charlotte, NC).
