Can't. Such equipment must have a minimum of 1600 watts. The power supply is a very important factor. For the safety of premises and equipment power supply should not operate at the limit of its capacity. There should always be a reserve power. If you are going to use the dual mining you will need a more powerful power supply.
180w per 1080 x 6 = 1080w + 100W for a basic mining rig(celeron with no fuzz). It's barely enough and wouldn't recommend. Always keep in mind to have 80% full load of your PSU, so 1200 would be 960w.
I wouldn't try it. 1200W could be enough with undervolting, but it is too big risk. When you invest that much money in the cards I would also invest in a stronger PSU or 2 of them. Keep in mind that PSU is most efficient when running around 60%, meaning lower electricity bills
EDIT: Can mine ETH if other two won't be profitable for me sometimes - but I don't think it will be often, very rare I would say...
Mining ETH with GTX 1080 is obviously a bad idea, it's one of the worst cards for the Dagger algo.
Regarding your power question - yes, it is possible to run 6 cards off a good 1200W PSU, but they'll have to be tuned (undervolted) and you won't have much headroom left. I'm running my GTX 1080 cards ~ 160-170 W each, so 6 of them draw ~ 1000W. Totally normal 24/7 load for a high quality PSU like HX1200. But I still wouldn't do it in the long run, only temporary, if you absolutely have no other way. If a driver crashes for some reason your settings could reset to the defaults and if your miner automatically restarts you'll be mining at the stock voltage (and that will place your total power draw close to 1200W). And it's just not a good practice to run PSUs like that. They achieve max efficiency at ~ 50-70% load, and they're also cool and quiet in those modes. Again, any good PSU can run at max load 24/7 for a long time, but it's just not recommended. Similar to GPU temperatures: you could run your cards at ~ 90C, and they'll probably be fine, but most people prefer lower temps for longevity, power efficiency and just peace of mind.
What about 4 cards? can it safely run on 1200W PSU?