The best answer I can give you is: no, there is no LARGE pocket of air under frozen lake water. Usually when water freezes on the top of a lake, it will have the water underneath it to support the formation of the ice, so there is never an air pocket. If for some reason the lake drained (water was pumped out?) the ice would most likely collapse at a certain point because that situation may well cause a vacuum to form. However, there could be a small pocket of air from some strange rock abnormality at the surface, but on average, I'd say no. There are situations where methane gets caught under ice, but that is due to decomposition of organic material as a lake thaws.

There can be, but there doesn't have to be. Generally the way that this would happen is if the water level drops after the ice forms. This can happen if the lake is being drained by a stream.