A terrestrial planet in the kind of orbit that Gl581c is in would end up with a lot of tidally-generated activity. You'd probably end up with a thick carbon dioxide atmosphere from that activity, rather like Venus. (In fact, the atmosphere of Venus at the surface is above the critical point of carbon dioxide, so we could regard Venus as having a supercritical CO2 ocean!)
Remember that water vapour is a greenhouse gas, so an "ice planet" would be a very hot one! If the planet is icy (water-rich), the likely situation is a supercritical "ocean" over a high pressure ice mantle. The atmosphere may well contain some free oxygen (due to photodissociation of water by starlight) as well as things like nitrogen, water vapour and carbon dioxide. You might get liquid water in the form of rain at the top of the atmosphere, particularly over the nightside, but it would not reach the surface. This might constitute a "habitable region" of sorts, insert argument about whether abiogenesis mid-air in raindrops is possible...