All off those are problems on your end.
When you fly in for a really long time, there is two options: ether your consol is having hard time getting the right information from your hard-drive or your connection is slow at that moment and you are having trouble establishing solid connection to the host.
When you load into a black screen it usually means that you are being re-connected.
And the lag is caused by your or the hosts connection. ( Destiny is a peer2peer game and doesn't use servers for hosting pvp or pve)
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I'm having issues playing with my wife. It kicks one or both if us from parties too frequently to enjoy the game properly.