As far as I can tell, DirecTV Stream is an app that appears on your Roku, FireTV, TV or possibly other streaming devices, or you can also use it in a web browser after logging in. Therefore the capabilities depend to some extent on which device or platform you are using. I don't know what "double play" is but the app does have search functionality. I think Pandora would be handled by a separate app, but I could be wrong. I do believe it allows you to pause live TV but I am not certain how big the buffer is, that probably depends on the memory available in your device. I will note that in very limited testing the app seemed to work best on a FireTV; on at least one of the others it crashed a lot but I don't recall which one offhand. As for whole home functionality, my understanding is you can have many streams (up to 20 I think?) going at once but it is way more than an average home would have (unless you are wealthy and have a mansion, perhaps), only thing is you would need another device at each TV but such devices are pretty inexpensive nowadays. They do check what IP address the streams are going to so you can't have one account and use it for all your far-flung friends and relatives, however if you have a cottage or vacation home I think they do allow one or two streams to go there. You might run out of Internet bandwidth before you exceed the number of allowed streams in some situations.
One thing it can do that a sat receiver can't is let you watch a show "on demand" but the ability to do that (and the length of time you have to do it after the show has aired) may vary from channel to channel. So if you tried to pause a live show and ran out of memory, there's still some chance you could go back and watch it on demand, however you'd be subject to all the commercials again which you may or may not be able to skip.
For better answers you'd need to talk to someone who's had a lot more experience with the service.
Honestly my biggest complaint about services that use apps like this is that they appear to do a rush job to get them out the door, so to speak, and depending on the platform they can be really buggy and have memory leaks or crash a lot, and the developers never seem to have the time or interest to go back and fix such issues even if a bunch of users are complaining about them.