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.