Not always. It P2P but it's still Bungie's servers connecting players to a host. If the servers aren't stable then the game is going to suffer as exemplified by the hike in Beaver, Cabbage, Bee and other error codes when Bungie does work on the servers.
If the servers weren't necessary to play Destiny then there would never be downtime when they were being updated.
Correct. But you'll know when it's the actual bungie server host because when that stops working you don't get laggy players, you get DC'd and get an error code that explicitly says "something went wrong talking to the server"
99.9% of the time if you've got laggy players, it's their connection that's the problem.
It can happen both ways. Sometimes my ISP is acting up and I'll see player walking aimlessly, teleporting, running along the floor of the map under a kill zone and other times Bungie's servers are messing up and I see the same thing.
The only way to know the difference is to check the BPS on your ISP's connection while its happening. If your ISP is good at that moment, then it's gonna be a Bungie problem