Firstly, does Zurich airport allow you to sleep over night in the departure area? (My layover is the middle of winter so I don't want to be sleeping outside.)
If not, what is my cheapest option for off site accommodation?
If you don't have access to frequent traveller lounges, and you don't want to sleep in a chair, you can buy a ticket for one of the day rooms offered by the airport, which will cost you CHF84.
sleepinginairports.com also recommends the Starbucks at the airport, if you don't really need a bed.
In the city, which is easily reached by public transport, the youth hostel (40 min from the airport) is a reasonably priced option; how much you pay depends on how many people you're willing to share your room with.
The Zurich airport does not close overnight. Even though there are no flights overnight, you can generally stay there and sleep on the chairs. Even though it is not really comfortable it is more or less safe. It could be that you will be woken up by security, but they will only very rarely tell you that you have to move on. If they do so, just go around the corner to the next chairs. But this really happens only very rarely. If you really sleep in the departure area, it is also very safe, because people without a plane ticket won't be able to get in there. You could also sleep outside of the departure area in the public space of the airport. This is generally also very safe, however it could be that you're woken up by drunk party people that wait for an early train since there are some night clubs right next to the airport.
If you want to sleep outside you have several options. You could get one of the airport hotels located in Kloten. There is a bus that will you pick up there. You could also go to the city, especially Oerlikon or the area about the main station are located quite close to the airport. However, all these options aren't really cheap since Zurich is an expensive city.
It's certainly possible to stay overnight as the airport doesn't close. However, reviews online are that it may be hard to sleep. Take this from sleepinginairports:
But as soon as the main lights went out, we became surrounded by some very questionable people. Two old homeless men on one side who had cans of bourbon, were snoring loudly then yelling at one another for snoring, and at one stage started harassing other young travelers trying to sleep there, grabbing them and yelling - and security was no where to be seen. On the other side, a homeless woman who smelled strongly of vomit. In the end, we gave up our great couches and went to look for another spot.
That, however, was next to one of the Starbucks. Presumably if you go deeper into the airport, you may have better luck in finding a quiet spot:
We found another starbucks closer to the check in area that also had couches - they weren't as comfortable as the first ones, but we slept so much better! There weren't any dodgy looking people around there either. I would recommend THAT Starbucks.
There is Wifi but it costs, apparently.
While others have mentioned hostels or hotels, there is another option for you at the airport - sleep rooms! They're pricey, but for the novelty could be worth trying ;)
There are sleep rooms in transit area D. Single Rates: CHF 49 (3 hours), CHF 63 (6 hours), CHF 79 (more than 6 hours). Double Rates: CHF 75 (3 hours), CHF 94 (6 hours), CHF 119 (more than 6 hours). There is a shower facility.