NFL NETWORK

Sorry, nobody here knows if or when DirecTV Stream might carry NFL Network. 

Several years ago (back when the service was owned by AT&T and named DirecTV Now), they did carry NFL Network.  It was dropped in April 2019.  The reason reported at the time was that AT&T and the NFL couldn't negotiate an acceptable carriage price.  The NFL denied that, but when it comes to money, I don't really believe the NFL on anything.  At the time DirecTV Now was hemorrhaging subscribers as they raised their loss-leader pricing to something more viable, and clearly wouldn't absorb a significant hit from the cost of NFL Network without raising prices even more, which they clearly didn't want to do.  (Of course, we don't know the actual cost figures - those sorts of things are almost never disclosed - so we don't know whose story is truer.)

My personal guess is that NFL Network won't come back to DirecTV Stream unless it is on one of the highest (most expensive) tiers or (more likely) if they add an extra cost sports add-on.  But as @litzdog911 indicated, no one "out here" would know.