I'm wondering what the chances are that I call DirecTV and they actually know whether the receiver(s) they send me have serial or not.

Zero, I'm afraid. They can't really give you any details about the specific brands or models of equipment that an installer might show up with. Installation is done by someone "local" who is licensed. So if you call DirecTV and get everything set up, they call someone within a certain distance from you to come do the job.

However if you go straight to a local installer as I did, you might be able to get some information on different receivers and their features, and perhaps even request that they bring one or more of a certain model they may carry that is to your liking.

But if you're planning to do the job yourself, you can buy whichever model you prefer.