Did you mean to type "800 plus gold psu"? It looks like you meant to write 80+ gold, which is a certification level of efficiency. What's the actual power output of it? If it's actually 800w then no, you won't require a new one as the 1080 is a very efficient card.
You don't need a new monitor to buy a gtx 1080, but if you plan to game in native 4K then you obviously will..saying that though, one 1080 won't get you very smooth framerates in BF1 in my opinion, not at 4K.
I wish you'd taken the time to read your own post before posting, would make replying with advice a lot easier.