I don't recall Starwood or Hilton being listed in the UR website. My guess is that you will be booking rooms for them via a third party website (ex: Travelocity, Expedia, etc.). Those will NOT earn you those extra points from what I have seen so far. You can try contacting SPG/Hilton support but I remember seeing in their terms and conditions that you must reserve on their official website. See
here for more info from other FT members.
I didn't check, but if AA is listed on the UR Mall website, then yes, I don't see why your status and all the other perks won't follow that booking. As long as it redirects you to the official AA website, it shouldn't be an issue. You should get the extra point from Chase and receive all the benefits from AA. Booking through third party websites is a hit and miss at best. You can read from other threads here in FT where some members were able to get the points but there's no guarantee.
If you read the terms and conditions (AA, Hilton, etc.), they should have it buried somewhere in there mentioning this. Some even mention the sites that will not earn you extra points if you book through them.