Great question!
Just to review. A greenhouse gas is such because it traps radiation that is emitted/reflected off the earth by absorbing it. In this way, the energy (in the form of heat) is trapped in the atmosphere instead of being radiated to space. All the greenhouse gasses absorb some of this radiation that the earth is emitting. But different gasses absorb different wavelengths of radiation and with different strengths. Methane happens to absorb a portion of the infra red, which is a large part of the radiation emitted/reflected from earth. Further, it absorbs portions that other greenhouse gasses (CO₂ and H₂O) do not absorb. In this way, previously unabsorbed radiation is now absorbed, and the heat is trapped.

It would be great to look at the IR absorption spectra for each of these green house gasses. We could compare the absorption spectra to the spectra of IR radiation emitted by the earth's surface. Based on how these gasses absorb and what is available to be absorbed (emitted by earth) we can estimate the relative impact of each gas.

The overall impact of each gas is a function of the gas's concentration and absorption spectrum. You bring up a good point about the stability of the gasses. If they decompose, then their decomposition products will be present and may be harmful. For example methane might eventually decompose into its oxidized products, but I am not sure how long this takes.

These sites are a bit helpful, please let me know if anything is unclear! Thanks for the question!

The reason why some gases are stronger greenhouse gases than others is related to the strength of the bonds in the molecules, but it doesn't require the bonds to be broken.When light hits a molecule, it could either pass right by it, or be absorbed as energy. This isn't enough energy to break the bonds - it actually causes the molecules to vibrate. Different molecular compositions will be able to vibrate at different energies, and ones that can easily vibrate at energies comparable to the energies of infrared light will be strong greenhouse gases. The constituent atoms, bond strengths, and geometry of the molecule should all affect the spectrum of light that the molecule will react too.