It decomposes to CO2 and water vapor over about 10 years. The more methane we pump out, the longer decomp takes because there are fewer atmospheric hydroxyl radicals to drive the reaction.

The methane cycle I'm familiar with is related to things dying and getting buried underground or in the ocean. I don't think there is any comparable process to how photosynthesis consumes CO2. There are methanogen bacteria who consume it, but that's probably not a globally scalable solution. I think it would be cheaper just to capture and store it.