Ok, first off, this board isn't run by Ubisoft, or even affiliated with them. At the top of this board's page is a disclaimer which reads:
"Disclaimer"
"The Division Forum is not ran by Ubisoft or Massive Entertainment the makers of The Division. In no way shape or form will posting "fix your damn game" do anything. Posting glitches will also not help those glitches or disk issues to be fixed since we can not fix any of those. Please take those comments and or feedback to the actual Ubisoft forums. "
If you wish to speak with Ubisoft directly and report a bug, specifically for PS4, go to:
http://forums.ubi.com/forumdisplay.php/1125-PS4?s=534569883ae8a8b633b76aa73ca04f91
As for your weekly reward issue itself, we've had a few players here experiencing the same thing on the same system. As Ubisoft is unlikely to award gear for previous completions if they ever do recognize this as a bug or offer a fix, we would recommend trying the run with a 2nd character. We're not saying nuke the one with the bug, but to narrow down the issue, see if it's account wide or confined to only one agent, as your agents can trade any gear/weapons/etc through their stash box, even weekly awards. The root cause of this issue isn't fully understood at this time, so the more you can do to narrow it down, the faster they can get a fix going for yourself and others. I understand that this is just free labor as a tester for them, but it's better than just waiting.
Secondly, be aware that the weekly reward resets on Wednesday at 8pm est. If you begin running FL before this time, and finish after this time, the game will take that run as counting for the previous week, and may not correctly award loot. Personally, I believe this could be a trigger for the bug you and others are currently experiencing, but we won't know until Ubi says something officially.
In slightly better news, patch 1.2 should be incoming soon, and with it comes a LOT of bug fixes and tweaks. It may not be that comforting now, but it could correct your issue.
As to your delta issues, those may or may not be the games fault. Everyone has a perfect connection when they get asked about it, but do what you can to ensure that your connection isn't bogged down with other in house users or programs while playing. Just as with the PS3, ensure that you regularly delete messages and notifications that are no longer of consequence as well.
Crashes, on the other hand, can be the result of an overworked hard drive or other system components. The Division relies HEAVILY on client-side processing, much more so than many other games. A full or nearly full drive can lead to write or read errors, which in turn can cause an instability, leading to a crash. Overheating can also bring crashes, ensure that your system is properly ventilated, and check the vent ports for clogs regularly. The cooler it stays, the better it runs.
I'm sorry you're experiencing issues, (and damn frustrating ones at that) but as a fan run board, there's not much we can do but offer the solutions above, and continue to search for a cause as best we can to aid in a timely fix. If we hear ANYTHING about a resolution to this issue or a suspected cause officially from Ubisoft, we will dutifully inform fans here as best we can, and spread whatever information amounts to a fix as fast as possible, though I hope by then your issue resolves itself, or gets resolved by Ubisoft automatically.
Best of luck man.