When booking through AAdvantage Hotels (or other 3rd party such as Expedia), you will generally not receive points or nights from your hotel program. You should assume that you will not be earning elite points or nights.
You can try to add your hotel program to your reservation at check-in or by calling the property before arrival. Some users have reported earning points for incidentals, receiving basic welcome benefits, or earning nights (World of Hyatt members will receive benefits and earn points, for example).
It is ultimately the hotel property that will decide if you can use any part of your status benefits with a 3rd party booking.
This does not apply to bookings made through AAdvantage eShopping.