How to determine methane percentage at home?
A couple of ideas:


If you think the diluent gas is non-flammable, you should be able to use the adiabatic flame temperature of the gas when you burn it in air to estimate the composition.  You will need to carefully adjust your stove to give the optimal fuel/air blend (that maximizes the flame temperature).  Then you will need to accurately measure the flame temperature.  If you assume a heat capacity for the unknown diluent, you can estimate its concentration in the gas by comparing the measured temperature to the theoretical adiabatic flame temperature for pure propane burning in air.
Use a vacuum pump to empty a container of known volume, and then allow the cooking gas into the container at ~1 atm pressure.  Measure the density of the gas and compare to the density of pure propane.
If you think the diluent gas does not contain carbon: hook up flow meters to the gas bottle (and an air or oxygen stream) to form a torch flame with known gas flow rate.  Collect all of the the exhaust produced by the flame, pass it through a cold trap to freeze out water vapor, and then pass the cold flame exhaust over a bed of anhydrous sodium hydroxide of known initial mass.  It will absorb the carbon dioxide from the exhaust.  Measure the mass increase of the sodium hydroxide.  By comparing the moles of gas that flow into the flame to the moles of CO2 absorbed (as calculated from the mass increase) you can estimate gas purity.
Find a spectrometer and record the IR or Raman spectrum of the gas!  This will probably be more accurate than any of the other methods.