I think your fridge is working! After it ignites, it is silent. Do you have it on "auto?" If so, the only light that will light is the auto light. The auto setting will run the fridge on electricity if you are on hookups and on gas if you have no external power.
If you push the gas button, it will only run on gas, not on power.
If it is running on gas but there is a problem, the check light will come on. You may hear it periodically clicking, trying to light.
I don't have any suggestions about the shower diverter -- my trailer does not have one of those. Maybe try this -- loop a cable tie around the hose, not real tight. Then slip it over the diverter lever to hold it in place. Then take it off when you are done showering.